id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,964 | KP-4.88 | Kirby Problem 4.88 | Let $\tau: S^{4} \to S^{4}$ be a free (hence orientation-reversing) involution. Is there an embedded $S^{2} \subset S^{4}$ that is invariant under $\tau$? This is equivalent to the existence of an embedded $\mathbb{RP}^{2}$ in $S^{4}/\tau \cong \mathbb{RP}^{4}$, carrying the nontrivial class in $H_{2}(\mathbb{RP}^{4};\... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.88.
Literature notes:
(1) This is an instance of the codimension two splitting problem [CS74] and may be asked in the smooth or topological category (where one would be looking for an $\mathbb{RP}^{2}$ with a normal bundle). Anot... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,965 | KP-4.89 | Kirby Problem 4.89 | Classify smooth, effective circle actions on simply connected 4-manifolds with boundary.
(a) Classify simply connected4-manifolds with boundary that admit circle actions.
(b) For each such 4-manifold, classify the circle actions up to conjugation by diffeomorphisms.
(c) Given a 4-manifold X with a circle action on i... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.89.
Literature notes:
(1) Information about the smooth classification of the closed 4-manifolds admitting a circle action was obtained by Fintushel and Pao [FP77] via Pao’s replacement trick [Pao77]. (These papers assumed the tru... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,966 | KP-4.90 | Kirby Problem 4.90 | Do the Chern numbers $c^{2}_{1}$ and $c_{2}$ of every closed, symplectic 4–manifold X that is not a ruled surface satisfy the following?
(a) $c^{2}_{1} \leq 3c_{2}$.
(b) $c^{2}_{1} =3c_{2}$ >0 if and only if $X =\mathbb{CP}^{2}$ or X is a complex ball quotient.
(c) $c_{2} \geq$ 0. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.90.
Literature notes:
(1) These invariants depend only on the underlying homotopy type of the 4–manifold X, satisfying the identities $c^{2}_{1} =2\chi+3\sigma$ and $c_{2} =\chi$ where $\chi$ and $\sigma$ are the Euler characteri... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,967 | KP-4.91 | Kirby Problem 4.91 | Present a topological construction of symplectic fake projective planes. Does there exist a symplectic fake projective plane that is not a complex ball quotient? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.91.
Literature notes:
(1) A fake projective plane (FPP) is a closed 4–manifold X with the same rational cohomology ring as the complex projective plane $\mathbb{CP}^{2}$ but not diffeomorphic to it. By Yau’s solution of the Calab... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,968 | KP-4.92 | Kirby Problem 4.92 | Is every symplectic Calabi-Yau surface diffeomorphic to either the K3 surface, the Enriques surface or $a T^{2}-bundle$ over $T^{2}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.92.
Literature notes:
(1) Here a symplectic Calabi-Yau surface (SCY) is defined as a closed symplectic 4-manifold with torsion first Chern class.
(2) To date, the only smooth 4–manifolds known to support a symplectic structure w... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,969 | KP-4.93 | Kirby Problem 4.93 | Is every symplectic form on the standard K3 surface symplectomorphic to a Kähler form? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.93.
Literature notes:
(1) This question has several notable generalizations. First, since every symplectic form on the K3 surface is cohomologous to a Kähler form [Li08, 4.10], this is a special case of the following broader ques... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,970 | KP-4.94 | Kirby Problem 4.94 | Are homotopy equivalent Horikawa surfaces in different deformation classes diffeomorphic as 4–manifolds? Are they symplectomorphic? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.94.
Literature notes:
(1) AHorikawa surfaceis a minimal (non-singular) complex projective surface of general type whose Chern numbers satisfy $5c^{2}_{1} = c_{2}$ −36 (i.e. it lies on the Noether line), or equivalently, $c^{2}_{1... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,971 | KP-4.95 | Kirby Problem 4.95 | (a) Is there a closed hyperbolic oriented 4-manifold that admits a symplectic structure?
(b) Do the Seiberg–Witten invariants vanish on every closed hyperbolic 4manifold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.95.
Literature notes:
(1) In a preprint [Sto25], Stover announces the construction of a hyperbolic orbifold structure on $\mathbb{CP}^{2}$. This resolves an orbifold version of this problem.
(2) By work of Taubes [Tau94], a posi... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,972 | KP-4.96 | Kirby Problem 4.96 | Does there exist a pair of symplectic 4–manifolds $(X_{1}, \omega_{1})and (X_{2}, \omega_{2})$, where there is a diffeomorphism $f: X_{1} \to X_{2}$ such that $f*(c_{1}(X_{2}, \omega_{2})) = c_{1}(X_{1}, \omega_{1})$ and $f*([\omega_{2}]) = [\omega_{1}] \in H^{2}(X_{1},\mathbb{R})$, but $(X_{1}, \omega_{1})$ is not sym... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.96.
Literature notes:
(1) It may be difficult to detect such a difference with current invariants due to Taubes’ result relating Gromov–Witten invariants and Seiberg–Witten invariants [Tau96].
(2) There is also a relative versio... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2608.09361",
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **open**; update the literature note: the relative question of Remark (2) now has a claimed affirmative answer in an unmarked form (unrefereed single-author preprint). K3 (2026), Problem 4.96 (pp. 269–270, scribed by L. Starkston and J. Van Horn-Morris) asks for diffeo... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,973 | KP-4.97 | Kirby Problem 4.97 | Let $\lambda:=c^{2}_{1}/c_{2}$ be the Chern slope of a closed, almost complex 4–manifold X. What is the supremum of $\lambda as X$ ranges over the following families?
(a) Symplectic $\Sigma_{g}–bundles$ over $\Sigma_{h}$, with g, $h \geq$ 2.
(b) Holomorphic $\Sigma_{g}–bundles$ over $\Sigma_{h}$, with g, $h \geq$ 2.
... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.97.
Literature notes:
(1) Recall that the Chern numbers of X are determined by its homotopy type. Any $\Sigma_{g}–bundle$ over $\Sigma_{h}$ with $g \geq$ 2 or a Lefschetz fibration (with nonempty critical set) over $S^{2}$ can be... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,974 | KP-4.98 | Kirby Problem 4.98 | Does every closed symplectic 4-manifold admit inequivalent Lefschetz pencils with the same fiber genus g, for sufficiently large g? How about infinitely many? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.98.
Literature notes:
(1) Here Lefschetz pencils are assumed to have base points, whereas Lefschetz fibrations do not. Equivalence is defined by a diffeomorphism of the symplectic 4–manifold commuting with any pair of pencil/fibr... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2602.10051",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **partially_solved (ruled surfaces only, by an unrefereed preprint; open in general)**. K3 (2026), Problem 4.98 (pp. 271–272, proposed by İ. Baykur) asks whether every closed symplectic 4-manifold admits inequivalent Lefschetz pencils of the same fiber genus, for every suff... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,975 | KP-4.99 | Kirby Problem 4.99 | Let X be a closed symplectic 4-manifold. Let $T \subset X$ be a symplectic submanifold that is diffeomorphic to a 2-dimensional torus such that $[T]^{2} =$ 0. Let $X_{K}$ be a manifold obtained by Fintushel–Stern knot surgery on T using a knot $K \subset S^{3}$. If $X_{K}$ has a symplectic structure, must K be a fibere... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.99.
Literature notes:
(1) Fintushel and Stern [FS98] showed that $X_{K}$ has a symplectic structure if K is a fibered knot, and raised the question in the problem.
(2) One can ask the question for the particular case when X admi... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,976 | KP-4.100 | Kirby Problem 4.100 | Given a closed, connected, symplectic 4-manifold (X, $\omega)$ and $c \in H_{2}(X,\mathbb{Z})$ represented by an embedded, connected, oriented, smooth surface S such that
(i) $x[\omega]$, cy $>$ 0, and
(ii) $xc_{1}(X, \omega)$, cy $=\chi(S) +S^{2}$, is c represented by an embedded, connected, symplectic surface? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.100.
Literature notes:
(1) This is a special case of the question of which homology classes in a symplectic manifold are represented by symplectic surfaces. By the adjunction formula, the homology class of every embedded symplect... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,977 | KP-4.101 | Kirby Problem 4.101 | Is every smooth symplectic surface in $(\mathbb{CP}^{2}, \omega_{FS})$ symplectically isotopic to a complex curve? Equivalently, is there a unique symplectic isotopy class of smoothly embedded symplectic surfaces of degree d in $(\mathbb{CP}^{2}, \omega_{FS})$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.101.
Literature notes:
(1) Here $\omega_{FS}$ denotes the standard Fubini–Study symplectic form on $\mathbb{CP}^{2}$. A symplectic surface in $\mathbb{CP}^{2}$ is said to have degree d if it represents the homology class $dh\in H... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,978 | KP-4.102 | Kirby Problem 4.102 | Is every symplectic rational cuspidal curve in $(\mathbb{CP}^{2}, \omega_{FS})$ equisingularly symplectically isotopic to a complex curve? More generally, which types of singular symplectic surfaces in $(\mathbb{CP}^{2}, \omega_{FS})$ (where type is specified by the genus of each irreducible component and the topologic... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.102.
Literature notes:
(1) This is a singular version of the symplectic isotopy problem (Problem 4.101). Note that “curve” here refers to a real 2-dimensional surface in analogy with the terminology for complex/pseudoholomorphic ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,979 | KP-4.103 | Kirby Problem 4.103 | (a) What polynomials can occur as the Alexander polynomials of complex plane algebraic curves?
(b) More generally, what are the conditions that must be satisfied by a finitely presented group G, so that there exist a plane algebraic curve having Gas the fundamental group of its complement? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.103.
Literature notes:
(1) Let C be an algebraic curve in $\mathbb{C}^{2}$. The Alexander polynomial of C relative to a surjection $\pi_{1}(\mathbb{C}^{2} \setminus$ C) $\to \mathbb{Z}$ is the characteristic polynomial the automo... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,980 | KP-4.104 | Kirby Problem 4.104 | Does there exist a transverse link $L \subset (S^{3}, \xi_{std})$ bounding a pair of complex curves in $B^{4} \subset \mathbb{C}^{2}$ that are isotopic through embedded smooth surfaces but not through complex curves? Smoothly isotopic but not symplectically? Can there be infinitely many of them? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.104.
Literature notes:
(1) By Rudolph [Rud83] and Boileau-Orevkov [BO01], L should be a quasipositive link and the complex curves can be realized as quasipositive surfaces in $B^{4}$. The boundary braids of the latter do not nece... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": null,
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **open**: the known pairs of complex curves with the same boundary (Baykur–Van Horn-Morris; Hayden) are not smoothly isotopic, and the question asks for smoothly isotopic ones. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,981 | KP-4.105 | Kirby Problem 4.105 | Does there exist a planar contact 3-manifold that has infinitely many distinct Stein fillings? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.105.
Literature notes:
(1) A contact 3-manifold is planar if it is supported by a planar open book decomposition.
(2) There are many interesting classes of planar contact 3-manifolds, for example lens spaces and links of rationa... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2607.21303",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **solved (yes; unrefereed preprint)**. K3 (2026), Problem 4.105 (p. 278, scribed by J. Van Horn-Morris) asks whether some planar contact 3-manifold has infinitely many distinct Stein fillings. R. İ. Baykur, [arXiv:2607.21303](https://arxiv.org/abs/2607.21303) (July 2026), T... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,982 | KP-4.106 | Kirby Problem 4.106 | Is the exact symplectomorphism type of $T*X^{4}$ sensitive to the smooth structure on a 4-manifold X, or does it depend only on the simple-homotopy or homeomorphism type of X? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.106.
Literature notes:
(1) Recall that the cotangent bundle $T^{*}X$ has a canonical 1-form $\lambda_{can}$; the canonical symplectic form on $T^{*}X$ isd $\lambda_{can}. A$ diffeomorphism $\varphi: T^{*}X \to T^{*}Y$ is an exact... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,983 | KP-4.107 | Kirby Problem 4.107 | Problems about contact hypersurfaces:
(a) Let (Y, $\xi)$ be a contact manifold and $(\mathbb{R} \times Y, \omega)$ its symplectization. Let f:Y $\to \mathbb{R} \times Y$ be a smooth embedding such thatf induces an isomorphism on homology. When can we isotope f so that its image is a contact type hypersurface?
(b) Spe... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.107.
Literature notes:
(1) There are some geometric obstructions to a hypersurface Y being isotopic to a contact type hypersurface in the ambient symplectic 4–manifold (M, $\omega)$; e.g. [Sul76] provides a necessary condition fo... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,984 | KP-4.108 | Kirby Problem 4.108 | Let $W_{+}$ and $W_{-}$ be two 4-dimensional Liouville domains with a contactomorphism $\Phi :\partial W_{-} \cong \partial W_{+}$. This determines an $\mathbb{R}-invariant$ contact structure $\xi$ on $\mathbb{R} \times X$ where X is the gluing $X =W_{+} \cup _{\Phi}W_{-}$ Moreover, 0 $\times X \subset \mathbb{R} \time... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.108.
Literature notes:
(1) A hypersurface $\Sigma \subset Y$ of a contact manifold (Y, $\xi)$ is convex if there is a contact vector-field V that is transverse to $\Sigma. A$ contact manifold is overtwisted if there is an embedde... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,985 | KP-4.109 | Kirby Problem 4.109 | If $\Sigma \subset$ (X, $\omega)$ is a symplectic surface in a closed symplectic 4-manifold with $[\Sigma] = P D(k[\omega]), k \in \mathbb{Z}$, does (X $\setminus \Sigma, \omega)$ support a Weinstein structure? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.109.
Literature notes:
(1) In particular, one could ask this for surfaces in $(\mathbb{CP}^{2}, \omega_{std}). A$ negative answer to the question in this case would yield a counterexample to the symplectic isotopy problem (Proble... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://doi.org/10.4007/annals.2005.161.959",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**; update the literature note: the case singled out in Remark (1) is known in low degree. K3 (2026), Problem 4.109 (p. 281, scribed by L. Starkston and J. Van Horn-Morris) asks whether the complement of a symplectic surface Σ with [Σ] = PD(k[ω]) in a... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,986 | KP-4.110 | Kirby Problem 4.110 | Does there exist a 2-handlebody W that admits an exact symplectic structure with convex contact boundary that does not admit a Weinstein structure filling the same contact boundary? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.110.
Literature notes:
(1) So far, all known examples of exact symplectic manifolds that we can obstruct from having a Weinstein structure are known to have handle structures requiring 3-handles [Mc D91, Bow12].
(2) We have limi... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": null,
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **open**: every known exact filling that is not Weinstein (McDuff; Bowden) needs 3-handles, and the question asks for a 2-handlebody. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,987 | KP-4.111 | Kirby Problem 4.111 | Is trisection genus additive? In other words, must it be the case that $g(X\#X1)$ =g(X) +g(X1). | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.111.
Literature notes:
(1) This is Conjecture 1.6 of [LCM22].
(2) The trisection genus of a 4–manifold X is $g(X)$ =min\{g|X admits a genus g trisection\}.
(3) $If\mathfrak{T}and\mathfrak{T}1$ are trisections for X and X1 of ge... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,988 | KP-4.112 | Kirby Problem 4.112 | Is every trisection of the 4-sphere with positive genus a stabilization of the genus zero trisection? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.112.
Literature notes:
(1) This is Conjecture 3.11 of [MSZ16]. The analogous result regarding Heegaard splittings of the 3–sphere is Waldhausen’s Theorem [Wal68a]. Connections between this problem, the Generalized Property R Conj... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://doi.org/10.1090/proc/13105",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**; update the literature note with the proved cases. J. Meier (K3 2026, Problem 4.112, p. 283) asks whether every positive-genus trisection of S⁴ is a stabilization of the genus-zero one (Conjecture 3.11 of Meier–Schirmer–Zupan). The same paper, [Pro... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,989 | KP-4.113 | Kirby Problem 4.113 | Which closed, oriented, smooth 4–manifolds admit genus–3 trisections? Which ones admit genus–3 simplified trisections? How about genus–4? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.113.
Literature notes:
(1) There is a complete classification of trisections of genus $g \leq$ 2 and only a handful of standard 4–manifolds admit them: $S^{4}$ for g =0; $\mathbb{CP}^{2}, \mathbb{CP}^{2}$, or $S^{1} \times S^{3}$... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,990 | KP-4.114 | Kirby Problem 4.114 | For a given Heegaard splitting of a closed orientable3–manifold, classify self-indexing Morse functions that give the given Heegaard splitting, up to $C\infty$ right-left (or right) equivalence. Likewise, for a given (simplified) trisection of a closed orientable 4–manifold, classify generic maps that give the given tr... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.114.
Literature notes:
(1) Let $f_{i}: M_{i} \to N_{i}$ be $C\infty$ maps between smooth manifolds, $i =$ 0,1. We say that they are $C\infty$ right-left equivalent if there exist diffeomorphisms $\phi: M_{0} \to M_{1}$ and $\psi:... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,991 | KP-4.115 | Kirby Problem 4.115 | (a) Find two diffeomorphic but non-isotopic trisections of the same4–manifold.
(b) Find two non-diffeomorphic balanced trisections of the same genus on a closed simply connected 4–manifold. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.115.
Literature notes:
(1) Part of the difficulty of Problem (a)is that it is connected to the problem of understanding the smooth mapping class groups of 4–manifolds. In particular, if a 4–manifold X has trivial smooth mapping c... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,992 | KP-4.116 | Kirby Problem 4.116 | Is there an algorithm to compute ‘distance’ in the cut complex of a trisection? Is the L-invariant computable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.116.
Literature notes:
(1) A Heegaard splitting for a 3-manifold M is defined by two “cut systems” of curves on a Heegaard surface $\Sigma$. Various notions of ‘distance’ (e.g. the Hempel distance) for Heegaard splittings have be... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2608.27811",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **partially_solved (second question: no, by an unrefereed preprint; the first is open)**. K3 (2026), Problem 4.116 (pp. 286–287, proposed by A. Thompson) asks whether 'distance' in the cut complex of a given trisection can be computed, and whether the Kirby–Thompson L-invar... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,993 | KP-4.117 | Kirby Problem 4.117 | Let X be a closed, orientable, smooth 4-manifold, with $g(X)$ the trisection genus of X. Does $g(X) =\chi(X) -2+3rk(\pi_{1}(X))$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.117.
Literature notes:
(1) Chu and Tillmann proved that $\chi(X) -2+3rk(\pi_{1}(X))$ is a lower bound for $g(X)$ [CT19]. It is reasonable to expect there to exist manifolds for which this inequality is strict, but at present, Chu... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,994 | KP-4.118 | Kirby Problem 4.118 | Does every simply connected, closed, smooth 4-manifold admit a handle decomposition without any 1-handles? Without 1-handles and 3-handles? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.118.
Literature notes:
(1) A manifold is called geometrically simply connected if it admits a handle decomposition without 1-handles. Any simply connected closed manifold of any dimension other than four is geometrically simply c... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,995 | KP-4.119 | Kirby Problem 4.119 | Is every topological 4-manifold homeomorphic to a CW complex? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.119.
Literature notes:
(1) It follows from the work of Kirby and Siebenmann [KS77] that every topological manifold M has the homotopy type of a CW complex, and moreover there is a canonical simple homotopy type of CW complexes ho... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,996 | KP-4.120 | Kirby Problem 4.120 | Which closed, smooth 4–manifolds admit achiral Lefschetz pencils? Does every simply connected 4–manifold have one? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.120.
Literature notes:
(1) Achiral Lefschetz pencils are generalizations of Lefschetz pencils, where the local models for nodal singularities and base points are allowed to reverse orientations. So, they are also defined on nonor... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,997 | KP-4.121 | Kirby Problem 4.121 | Which closed, smooth4–manifolds admit open book decompositions? In particular, does every closed, simply connected 4–manifold with signature zero admit one? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.121.
Literature notes:
(1) An open book decomposition of an n-dimensional manifold X is given by a smooth fibration f: $X \setminus L \to S^{1}$, where the binding $L \subset X$ is an (n−2)–dimensional embedded submanifold with a... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,998 | KP-4.122 | Kirby Problem 4.122 | Is there a universal branching surface $S \subset S^{4}$ such that every closed, orientable 4-manifold W admits a branched covering $W \to S^{4}$ with branchin g locus S? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.122.
Literature notes:
(1) This question originated in the introduction of [PZ05] by Piergallini– Zuddas.
(2) Using the signature, it can be shown that a universal branching surface in $S^{4}$ is necessarily disconnected [Vir84,... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,999 | KP-4.123 | Kirby Problem 4.123 | Is every closed leaf of a two dimensional co-orientable smooth taut foliation of an oriented 4-manifold smoothly genus-minimizing in its homology class? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.123.
Literature notes:
(1) This question is due to Kronheimer [Kro98, Question 7.12], and is the natural generalization of the corresponding three-dimensional result of Thurston [Thu86]. Here we say that the $foliation\mathcal{F}... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,000 | KP-4.124 | Kirby Problem 4.124 | Does there exist a hyperbolic integer homology four-sphere? What about an arithmetic one? Homology four-spheres have Euler characteristic 2, so it makes sense to ask more generally if there exist any closed hyperbolic fourmanifold with Euler characteristic 2. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.124.
Literature notes:
(1) If any closed, hyperbolic manifold with Euler characteristic 2 exists, there are only finitely many. This is because the Gauss-Bonnet Theorem says that $V ol(M) =^{4}_{3}\pi^{2}\chi(M)$ for a hyperbolic... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": null,
"checked_at": "2026-10-04",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-04] **open**: no closed hyperbolic 4-manifold with χ = 2 is known and none is excluded; the theorem of Belolipetsky quoted in the remarks only says where an arithmetic one would have to arise (as a torsion-free subgroup of index 28800 in the Coxeter group [5,3,3,3]). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,001 | KP-4.125 | Kirby Problem 4.125 | Is there a noncompact, finite volume, orientable hyperbolic four-manifold without a spin structure? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.125.
Literature notes:
All compact orientable manifolds with dimension at most three admit spin structures. Sullivan observed that every finite volume hyperbolic nmanifold has a finite cover that admits a spin structure [Sul79b, ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: Riolo--Rizzi construct a noncompact finite-volume orientable hyperbolic 4-manifold without any spin structure, exactly answering the question affirmatively. This is a consistency correction based on that cited review; the discussion does not independently re... | [
{
"url": "https://arxiv.org/abs/2510.12657",
"label": "Stefano Riolo and Edoardo Rizzi, A cusped hyperbolic 4-manifold without spin structures, arXiv:2510.12657 (2025)."
},
{
"url": "https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf",
"label": "R. I. Bayk... | 2026-09-27T00:00:00 | Stefano Riolo and Edoardo Rizzi, A cusped hyperbolic 4-manifold without spin structures, arXiv:2510.12657 (2025).: https://arxiv.org/abs/2510.12657
R. I. Baykur, R. C. Kirby, and D. Ruberman (eds.), K3, AMS 295 (2026), Problem 4.125.: https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pd... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,002 | KP-4.126 | Kirby Problem 4.126 | (a) If M is a closed, orientable hyperbolic 4-manifold then it always has signature 0, because its Pontryagin class vanishes [Che55]. This implies that M bounds a compact, orientable 5-manifold. Is M always a geometric boundary?
(b) For general n, suppose that M is a closed hyperbolic n-manifold. Is M a geometric boun... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.126.
Literature notes:
(1) Following [LR00], we say that $M^{n}$ is a geometric boundary if it is the totally geodesic boundary of a compact hyperbolic manifold $W^{n}+^{1}$.
(2) Perhaps a good example to start with for(a)is the... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,003 | KP-4.127 | Kirby Problem 4.127 | Given an aspherical closed (or compact and bounded by flat 3-manifolds) 4-manifold M and a self-diffeomorphism f of M, find necessary and sufficient conditions on f so that the resulting 5-dimensional mapping torus $M_{f}$ admits a hyperbolic structure. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.127.
Literature notes:
(1) By hyperbolization, in dimension two the sufficient and necessary condition is that f is pseudo-Anosov (see Thurston [Thu82, Thu22] and Otal [Ota01] for a complete proof). There are some analogues in st... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,004 | KP-4.128 | Kirby Problem 4.128 | What is the structure of 4-manifolds that admit a Riemannian metric of positive scalar curvature? There are variations of this problem for different classes of manifolds.
(a) Is every closed simply connected PSC 4-manifold diffeomorphic to a connected sum of copies of $\mathbb{CP}^{2}, \mathbb{CP}^{2}$, and $S^{2} \ti... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.128.
Literature notes:
(1) Let us say that a smooth manifold is a PSC manifold if it admits a Riemannian metric of positive scalar curvature.
(2) The corresponding problem in dimension 2 is easy by the Gauss-Bonnet theorem, and ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2609.37948",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**; update the literature note: the test case in Remark (6) is now answered (no; unrefereed preprint), and **fix the statement**. K3 (2026), Problem 4.128 (pp. 295–296, scribed by D. Ruberman) asks for the structure of 4-manifolds with positive scalar... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,005 | KP-4.129 | Kirby Problem 4.129 | Given a closed, 4-dimensional PSC manifold M, is there a (possibly disconnected) 4-dimensional orbifold $M^{1}$ with isolated singularities such that the following hold. (I) The orbifold M1 also admits a metric of positive scalar curvature. (II) All components of $M^{1}$ have finite orbifold-fundamental group. (III) M ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.129.
Literature notes:
(1) Because the orbifold $M^{1}$ has isolated singularities, its orbifold fundamental group is just the fundamental group of its regular part.
(2) The following converse statement is true due to the work o... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,006 | KP-4.130 | Kirby Problem 4.130 | Does longitudinal knot surgery using a knot K, along a fiber in a K3 surface always yield a reducible 4–manifold? A completely decomposable 4–manifold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.130.
Literature notes:
(1) Longitudinal knot surgeryis a variation of Fintushel-Stern knot surgery [FS98] with a different gluing map. The original Fintushel-Stern version is defined using a knot K and a square-0 torus T in a 4-m... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,007 | KP-4.131 | Kirby Problem 4.131 | Does every Lipschitz 4-manifold admit a smooth structure? Is this smooth structure unique if so? Some more specific, related questions are as follows.
(a) Is there a topological, spin, closed, indefinite 4-manifold X that admits a Lipschitz structure and violates the 10/8-inequality $b_{2}(X) \geq 5|\sigma(X)|$ +2 by ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.131.
Literature notes:
(1) In dimension $n \ne$ 4, every topological n-manifold X admits a Lipschitz structure by a theorem of Sullivan [Sul79b]. On the other hand, Donaldson and Sullivan [DS89, Theorem 2] showed that there exist... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://doi.org/10.4310/MRL.2001.v8.n3.a5",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**, but **fix part (a)**: the fractions are lost. K3 (2026), Problem 4.131 (pp. 298–299, proposed by J. Chaidez, D. Gabai and H. Konno) asks whether every Lipschitz 4-manifold admits a smooth structure, and a unique one; part (a) asks for a closed, sp... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,008 | KP-5.1 | Kirby Problem 5.1 | Does every cellular set in the plane have the fixed point property? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.1.
Literature notes:
- A space has the fixed point property (FPP) if every self-map has a fixed point. A set in an $n$-manifold is cellular if it is the intersection of a countable sequence of embedded closed $n$-cells, each cont... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,009 | KP-5.2 | Kirby Problem 5.2 | (Doubly-Small Morphisms of Manifolds).
- Suppose that $h: \R^{n} \to \R^{n}$ is a homeomorphism (or diffeomorphism) which satisfies two smallness hypotheses:
- every orbit of $h$ (of any $x\in \R^{n}$, under all powers of $h$) is uniformly bounded in diameter (by 1 say), and
- some subsequence of powers of $h$ conve... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.2.
Literature notes:
- The case $n=1$ is trivial, $n=2$ seems likely to be true, as a consequence of results of Brouwer and Cartwright-Littlewood (nicely and succinctly reproved in [Bro84] and [Bro77]), and $n\geq 3$ is open.
- ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | [Discussion #4 literature review, 2026-10-09] . The record's note says n = 2 "seems likely to be true"; it is in print. Closed disc: L. G. Oversteegen, E. D. Tymchatyn, Proc. Amer. Math. Soc. 110 (1990), 1083–1088, Theorem 1 (the abstract says that the result answers a question of R. D. Edwards, who is also the propose... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,010 | KP-5.3 | Kirby Problem 5.3 | (Hilbert--Smith Conjecture).
- The Hilbert--Smith Conjecture [Smi41] asserts that a locally compact subgroup of the homeomorphism group of a connected manifold must be a Lie group.
- Conjecture: The free-action subset of any Cantor group action on an ENR is a homology-$Z$-subset (of the ENR). | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.3.
Literature notes:
- We first define the terms in Conjecture (b).
- A Cantor group is a topological group $G$ whose underlying space is a Cantor set (space). That is, $G$ is a profinite group that is non-finite and 2nd countab... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,011 | KP-5.4 | Kirby Problem 5.4 | Is the homeomorphism group of a manifold an absolute neighborhood retract (ANR)? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.4.
Literature notes:
- This is [Kir97, Problem 5.27] and is also listed in [Wes90]. In dimension 2 it was shown to hold by Mason [Mas71] and Luke-Mason [LM72]; it is unknown for manifolds of dimension greater than 2.
- It is not... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,012 | KP-5.5 | Kirby Problem 5.5 | Is the connected sum of (locally flat) pairs
$$
(M^{n+k}_{1},N^{n}_{1})\#(M^{n+k}_{2},N^{n}_{2})
$$
well-defined in the topological category? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.5.
Literature notes:
When $k=1$, this is easily true from Brown's paper showing that locally flat codimension one submanifolds are flat [Bro62]. According to [Liv24], it is well-defined in any dimension when $k=2$. This uses the ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,013 | KP-5.6 | Kirby Problem 5.6 | - Does every closed, PL, orientable $n$-manifold admit an $n$-fold branched covering map over $S^{n}$?
Assuming that the answer to this problem is "yes", the following follow-up questions would be natural to ask.
- Does every $n$-manifold admit an $n$-fold branched covering over $S^{n}$ with branch locus a codimensio... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.6.
Literature notes:
- A classical theorem of Alexander [Ale20] says that every $n$-manifold admits a branched covering map over $S^{n}$ with no restriction on degree of the covering. An easier version of this problem would be to... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,014 | KP-5.7 | Kirby Problem 5.7 | (Montgomery--Yang problem). Does there exist a pseudo-free, smooth, $S^{1}$ action on $S^{5}$ with more than three multiple orbits? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.7.
Literature notes:
- This is Problem 4.123 in [Kir97]; see the surveys by Kollár [Kol08] and Şavk [Şav24]. Recall that an $S^{1}$ action is pseudo-free if there are no fixed points and the orbits of finite isotropy are isolated... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,015 | KP-5.8 | Kirby Problem 5.8 | Is there a closed aspherical 5-manifold that is not triangulable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.8.
Literature notes:
- In this context, `triangulable' means homeomorphic to a simplicial complex. Davis and Januszkiewicz [DJ91] constructed aspherical non-triangulable 4-manifolds, by applying the process of `hyperbolization' t... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | [Discussion #4 literature review, 2026-10-08] , `KP-5.8` (3015): `10.1090/jams/829` → `10.1090/jams829`. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,016 | KP-5.9 | Kirby Problem 5.9 | Let $M_{1}$ and $M_{2}$ be smooth manifolds of dimension $n$. Suppose $M_{1}$ admits an $S$-map into $\R^{p}$. If $M_{2}$ is homeomorphic to $M_{1}$, then does $M_{2}$ admit an $S$-map as well? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.9.
Literature notes:
- For this problem, we consider $C^{\infty}$ maps $f:M\to \R^{p}$ of smooth $n$-dimensional manifolds into $\R^{p}$ with $n\geq p\geq 1$. Let $S$ be a set of (equivalence classes of) singularities of smooth m... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: The universal assertion is false. For S consisting of definite fold singularities, S-maps are special generic maps, and homeomorphic smooth manifolds can differ in whether they admit such maps; standard and suitable exotic spheres provide counterexamples. Th... | [
{
"url": "https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf",
"label": "R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.9, author's preliminary AMS version (2026)."
},
{
"url": "http... | 2026-09-27T00:00:00 | R. I. Baykur, R. C. Kirby, and D. Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, Problem 5.9, author's preliminary AMS version (2026).: https://bpb-us-e2.wpmucdn.com/websites.umass.edu/dist/b/22144/files/2026/04/K3-problem-list-watermarked.pdf
O. Saeki, Topology of special generic maps of manifolds int... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,017 | KP-5.10 | Kirby Problem 5.10 | The Andrews--Curtis Conjecture [AC65] for the trivial group: a presentation of the trivial group can be changed to the trivial presentation by Andrews--Curtis moves. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.10.
Literature notes:
- This problem appears in Problem 5.2 in [Kir97].
- Let $\mathcal{P}$ be a finite presentation of a given group $\pi$. The stabilized Andrews--Curtis moves (abbreviated AC moves) change the presentation $\m... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,018 | KP-5.11 | Kirby Problem 5.11 | (Whitehead's Asphericity Question). Is every subcomplex of an aspherical 2-complex aspherical? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.11.
Literature notes:
- This was problem 5.4 in [Kir97]; a thorough discussion of older progress on this problem, also known as the Whitehead Conjecture, may be found in [HAMS93, Chapter X]. It was stated as a question on page 42... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | [Discussion #4 literature review, 2026-10-08] `10.1142/$S^{1}$793525317500170` → `10.1142/S1793525317500170` (the corruption is already flagged in the note). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,019 | KP-5.12 | Kirby Problem 5.12 | (Zeeman Conjecture). If $K$ is a finite contractible 2-complex, then $K\times I$ collapses to a point [Zee64, Conjecture (1)]. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.12.
Literature notes:
This problem appears as part of Problem 5.2 in [Kir97].
The Zeeman Conjecture for a special polyhedron that is the spine of a compact 3-manifold is equivalent [GR83] to the Poincaré Conjecture, and thus tru... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,020 | KP-5.13 | Kirby Problem 5.13 | Let $M$ be a finite-volume hyperbolic $n$-manifold.
- Does it always have a finite cover with $b_{1}>0$?
- Does it always have a finite cover with fundamental group that surjects onto a non-abelian free group?
- Does it have a finite cover with cubulated fundamental group?
- When $n$ is odd, does it always have a f... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.13.
Literature notes:
- When $n=3$, these questions all have positive answers, by work of Agol [Ago13] and Wise [Wis21]. However, their methods break down when $n>3$. The crucial steps in their argument are as follows. By work of... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,021 | KP-5.14 | Kirby Problem 5.14 | Does there exist a 1-cusped finite-volume hyperbolic $n$-manifold for any $n\geq 5$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.14.
Literature notes:
- Hyperbolic $n$-manifolds with one cusp are abundant when $n=2$ or 3, and Kolpakov--Martelli [KM13a] constructed 1-cusped hyperbolic 4-manifolds. However, there are far fewer constructions of hyperbolic $n$... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,022 | KP-5.15 | Kirby Problem 5.15 | Suppose $M$ is a manifold with a complete Riemannian metric with nonnegative Ricci curvature. Is the fundamental group of $M$ finitely generated? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.15.
Literature notes:
- This is a conjecture of Milnor, following on his paper [Mil68a]. The conjecture was known to hold in dimension 2 by work of Cohn-Vossen [CV35] and in dimension 3 by work of Liu [Liu13]. Bruè--Naber--Semola... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: The universal finite-generation assertion is false: smooth complete nonnegative-Ricci manifolds with infinitely generated fundamental group exist in dimensions 7 and 6. This is a consistency correction based on that cited review; the discussion does not inde... | [
{
"url": "https://annals.math.princeton.edu/2025/201-1/p04",
"label": "Elia Bruè, Aaron Naber, and Daniele Semola, Fundamental groups and the Milnor conjecture, Ann. of Math. 201 (2025), 225--289."
},
{
"url": "https://arxiv.org/abs/2311.12155",
"label": "Elia Bruè, Aaron Naber, and Daniele Semo... | 2026-09-27T00:00:00 | Elia Bruè, Aaron Naber, and Daniele Semola, Fundamental groups and the Milnor conjecture, Ann. of Math. 201 (2025), 225--289.: https://annals.math.princeton.edu/2025/201-1/p04
Elia Bruè, Aaron Naber, and Daniele Semola, Six dimensional counterexample to the Milnor Conjecture, arXiv:2311.12155.: https://arxiv.org/abs/23... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,023 | KP-5.16 | Kirby Problem 5.16 | Let $R$ be $\Z$ or a field. Let $A$ and $B$ be differential graded algebras so that either:
\begin{itemize}
- As a graded algebra, $A$ (respectively $B$) is isomorphic to the free, non-commutative $R$-algebra on finitely many homogeneous generators $x_{i}$, i.e., to the tensor algebra on the free $R$-module generated ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.16.
Literature notes:
- This question is inspired by contact topology, where many invariants take the form of a finitely-generated differential graded algebras up to a notion of equivalence.
- The kinds of dgas described in the ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,024 | KP-5.17 | Kirby Problem 5.17 | Let $(W,\omega,V_{i},\phi_{i})$ be two Weinstein structures on a fixed symplectic manifold $(W,\omega)$ (or equivalently consider two Weinstein handle decompositions). Is there a Weinstein homotopy from $(W,\omega,V_{1},\phi_{1})$ to $(W,\omega,V_{2},\phi_{2})$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.17.
Literature notes:
% The PDF prints V_2 in both terms of this convex combination; retained as printed.
- The convex combination $V_{t}=(1-t)V_{2}+tV_{2}$ is a Liouville vector field positively transverse to the boundary, but ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,025 | KP-5.18 | Kirby Problem 5.18 | - In higher dimensions, find the `non-analytic' cohomology module $\HH^{*}_{?}$ analogous to the analytic lattice cohomology $\HH^{*}_{\mathrm{an}}$, and a natural functor $\HH^{*}_{\mathrm{an}}\to \HH^{*}_{?}$ connecting them.
- Define a version $\ECH^{*}$ of Embedded Contact Homology for isolated complex singularity... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-5.18.
Literature notes:
- As background, consider a complex normal surface singularity with a rational homology sphere link. The (topological) lattice cohomology $\HH^{*}_{\mathrm{top}}$, associated with the link (which is a plumbe... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,026 | OPG-3031 | trace inequality | Let $A,B$ be positive semidefinite, by Jensen's inequality, it is easy to see $[tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}}$, whenever $s>r>0$.
What about the $tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}}$, is it still valid? | Source: Open Problem Garden. Original node ID: 3031. URL: http://www.openproblemgarden.org/op/trace_inequality.
Source subject path: Algebra.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/trace_inequality
- Subject(s): Algebr... | 1 | solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
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"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": null,
"checked_at": "2026-09-29",
"kind": "resolution",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-09-29] **solved (no)**. The version with the power inside the trace, tr[(A^s+B^s)^{1/s}] ≤ tr[(A^r+B^r)^{1/r}], fails in general whenever 1 < r < s. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,027 | OPG-23298 | Elementary symmetric of a sum of matrices | Problem
Given a Matrix $A$, the $k$-th elementary symmetric function of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by- $k$ principal minors.
Find a closed expression for the $k$-th elementary symmetric function of a sum of N $n$-by- $n$ matrices, with $0\le N\le k\le n$ by using partitions. | Source: Open Problem Garden. Original node ID: 23298. URL: http://www.openproblemgarden.org/op/elementary_symmetric_of_a_sum_of_matrices.
Source subject path: Algebra.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/elementary_s... | 2 | partially_solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] . The closed expression is Amitsur's formula (S. A. Amitsur, [Linear Multilinear Algebra 8 (1980), 177–182](https://doi.org/10.1080/03081088008817315)); the paper cited in the note ([arXiv:math/0612464](https://arxiv.org/abs/math/0612464)) treats special cases and leaves th... | null | Reconciled with the existing literature review of 2026-08-17: Costas-Santos supplies general expressions for elementary symmetric functions of sums of matrices, including the partition/Newton--Girard framework requested in the source. This is a consistency correction based on that cited review; the discussion does not ... | [
{
"url": "https://arxiv.org/abs/math/0612464",
"label": "R. S. Costas-Santos, On the elementary symmetric functions of a sum of matrices, J. Algebra Number Theory Adv. 1 (2009), 99--112."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath disc... | 2026-09-27T00:00:00 | R. S. Costas-Santos, On the elementary symmetric functions of a sum of matrices, J. Algebra Number Theory Adv. 1 (2009), 99--112.: https://arxiv.org/abs/math/0612464 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,028 | OPG-37283 | Finite Lattice Representation Problem | Conjecture
There exists a finite lattice which is not the congruence lattice of a finite algebra. | Source: Open Problem Garden. Original node ID: 37283. URL: http://www.openproblemgarden.org/op/finite_congruence_lattice_problem.
Source subject path: Algebra.
Source importance: Outstanding ✭✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/finite_congr... | 3 | partially_solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | false | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 12,
"name": "opengarden",
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"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | Duplicate of canonical problem ALG-046 ('Finite Lattice Representation Problem') | [
"ALG-046"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,029 | OPG-48715 | Sub-atomic product of funcoids is a categorical product | Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
- Product morphism is defined similarly to the category of topological spaces.
- Product object is the sub-atomic product.
- Projections are sub-atomic projections.... | Source: Open Problem Garden. Original node ID: 48715. URL: http://www.openproblemgarden.org/op/sub_atomic_product_of_funcoids_is_a_categorical_product.
Source subject path: Algebra.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/... | 1 | partially_solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,030 | OPG-50149 | inverse of an integer matrix | Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all $\ge 2$. Suppose X is an m-by-n integer matrix $(m \le n)$. Consider the partitioned matrix M = [D X]. Obviously M ha... | Source: Open Problem Garden. Original node ID: 50149. URL: http://www.openproblemgarden.org/op/inverse_of_an_integer_matrix.
Source subject path: Algebra.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/inverse_of_an_integer_ma... | 1 | partially_solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | false | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | Non-research homework/calculus submission on Open Problem Garden | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: For any full-row-rank integer matrix M=[D X], an integer right inverse exists iff the gcd of its m by m minors is one, equivalently iff the Smith normal form has every nonzero invariant factor one. This is a consistency correction based on that cited review;... | [
{
"url": "https://encyclopediaofmath.org/wiki/Normal_form_%28for_matrices%29",
"label": "Encyclopedia of Mathematics, Normal form (for matrices), Smith normal form (accessed 2026-08-17)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath disc... | 2026-09-27T00:00:00 | Encyclopedia of Mathematics, Normal form (for matrices), Smith normal form (accessed 2026-08-17).: https://encyclopediaofmath.org/wiki/Normal_form_%28for_matrices%29 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,031 | OPG-57824 | Graphs of exact colorings | Conjecture For $c \geq m \geq 1$, let $P(c,m)$ be the statement that given any exact $c$-coloring of the edges of a complete countably infinite graph (that is, a coloring with $c$ colors all of which must be used at least once), there exists an exactly $m$-colored countably infinite complete subgraph. Then $P(c,m)$ is ... | Source: Open Problem Garden. Original node ID: 57824. URL: http://www.openproblemgarden.org/op/graphs_of_exact_colorings.
Source subject path: Algebra.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/graphs_of_exact_colorings
-... | 1 | partially_solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | false | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
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"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | Duplicate of canonical problem OPG-37229 | [
"OPG-37229"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,032 | OPG-60031 | Waring rank of determinant | Question What is the Waring rank of the determinant of a $d \times d$ generic matrix?
For simplicity say we work over the complex numbers. The $d \times d$ generic matrix is the matrix with entries $x_{i,j}$ for $1 \leq i,j \leq d$. Its determinant is a homogeneous form of degree $d$, in $d^2$ variables. If $F$ is a h... | Source: Open Problem Garden. Original node ID: 60031. URL: http://www.openproblemgarden.org/op/waring_rank_of_determinant.
Source subject path: Algebra.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/waring_rank_of_determinant... | 1 | partially_solved | null | null | 4 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | [Discussion #4 literature review, 2026-10-08] [arXiv:2301.06586](https://arxiv.org/abs/2301.06586) is by R. Houston, A. P. Goucher and N. Johnston (cited: Conner, Harper and Landsberg). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,033 | OPG-725 | $C^r$ Stability Conjecture | Conjecture Any $C^r$ structurally stable diffeomorphism is hyperbolic. | Source: Open Problem Garden. Original node ID: 725. URL: http://www.openproblemgarden.org/op/c_r_stability_conjecture.
Source subject path: Analysis.
Source importance: Outstanding ✭✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/c_r_stability_conjectu... | 3 | partially_solved | null | null | 8 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
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"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
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"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,034 | OPG-36697 | Invariant subspace problem | Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace? | Source: Open Problem Garden. Original node ID: 36697. URL: http://www.openproblemgarden.org/op/invariant_subspace_problem.
Source subject path: Analysis.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/invariant_subspace_problem... | 2 | open | null | null | 8 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
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"description": "Limits, continuity, calculus, and function theory.",
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"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
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"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,035 | OPG-36928 | Criterion for boundedness of power series | Question Give a necessary and sufficient criterion for the sequence $(a_n)$ so that the power series $\sum_{n=0}^{\infty} a_n x^n$ is bounded for all $x \in \mathbb{R}$. | Source: Open Problem Garden. Original node ID: 36928. URL: http://www.openproblemgarden.org/op/criterion_for_boundedness_of_power_series.
Source subject path: Analysis.
Source importance: Low ✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/criterion_for_... | 1 | partially_solved | null | null | 8 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
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"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,036 | OPG-37185 | Something like Picard for 1-forms | Conjecture Let $D$ be the open unit disk in the complex plane and let $U_1,\dots,U_n$ be open sets such that $\bigcup_{j=1}^nU_j=D\setminus\{0\}$. Suppose there are injective holomorphic functions $f_j: U_j \to \mathbb{C},$ $j=1,\ldots,n,$ such that for the differentials we have ${\rm d}f_j={\rm d}f_k$ on any intersect... | Source: Open Problem Garden. Original node ID: 37185. URL: http://www.openproblemgarden.org/op/something_like_picard_for_1_forms.
Source subject path: Analysis.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/something_like_pic... | 1 | partially_solved | null | null | 8 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "scope_correction",
"evidence": "scope_review"
} | null | null | null | null | null | The recorded counterexample uses disconnected covering sets. The connected-set version intended by the Picard-type question remains unresolved; clarify that hypothesis.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The cited partial result or literal o... | [
{
"url": "https://mathoverflow.net/questions/61882/meromorphic-1-form-and-picards-theorem",
"label": "MathOverflow question 61882, Meromorphic 1-form and Picard's theorem, expert answers and discussion, checked 17 August 2026."
},
{
"url": "https://doi.org/10.5802/aif.1675",
"label": "B. Elsner,... | 2026-09-27T00:00:00 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,037 | OPG-41335 | Inequality for square summable complex series | Conjecture For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2$$ | Source: Open Problem Garden. Original node ID: 41335. URL: http://www.openproblemgarden.org/op/inequality_for_square_summable_complex_series.
Source subject path: Analysis.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/inequa... | 1 | solved | null | null | 8 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 8,
"name": "analysis",
"display_name": "Analysis",
"description": "Limits, continuity, calculus, and function theory.",
"slug": "analysis",
"order_index": 8,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": null,
"checked_at": "2026-09-29",
"kind": "resolution",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-09-29] **solved**. The inequality follows from Cauchy–Schwarz on the classes {2^k(2l+1)}, and the constant 6/π² is sharp. This argument was given in an anonymous comment of 3 Nov 2013 on the problem page, and the poster acknowledged it in 2014. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,038 | OPG-426 | Long rainbow arithmetic progressions | For $k\in \mathbb{N}$ let $T_k$ denote the minimal number $t\in \mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of $\{ 1,2,\ldots,tn\}$ for every $n\in \mathbb{N}$
Conjecture For all $k\geq 3$, $T_k=\Theta (k^2)$. | Source: Open Problem Garden. Original node ID: 426. URL: http://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/long_rainbow... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: The conjecture T_k=Theta(k^2) is false: Geneson proves the matching lower bound Omega(k^2 log k), yielding the exact order T_k=Theta(k^2 log k). This is a consistency correction based on that cited review; the discussion does not independently re-verify its ... | [
{
"url": "https://arxiv.org/abs/2607.15116",
"label": "Jesse Geneson, The order of long rainbow arithmetic progressions, arXiv:2607.15116 (2026)."
},
{
"url": "https://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions",
"label": "Open Problem Garden, Long rainbow arithmetic progr... | 2026-09-27T00:00:00 | Jesse Geneson, The order of long rainbow arithmetic progressions, arXiv:2607.15116 (2026).: https://arxiv.org/abs/2607.15116
Open Problem Garden, Long rainbow arithmetic progressions (node 426).: https://www.openproblemgarden.org/op/long_rainbow_arithemtic_progressions | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,039 | OPG-478 | Rainbow AP(4) in an almost equinumerous coloring | Problem Do 4-colorings of $\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either $\lfloor p/4\rfloor$ or $\lceil p/4\rceil$? | Source: Open Problem Garden. Original node ID: 478. URL: http://www.openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/r... | 1 | open | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,040 | OPG-618 | Monotone 4-term Arithmetic Progressions | Question Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions? | Source: Open Problem Garden. Original node ID: 618. URL: http://www.openproblemgarden.org/op/monotone_4_term_arithmetic_progressions.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/monotone_... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,041 | OPG-636 | Even vs. odd latin squares | A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise.
Conjecture For every positive even integer $n$, the number of even latin squares of order $n$ and the number of odd latin squares of order $n$ are different. | Source: Open Problem Garden. Original node ID: 636. URL: http://www.openproblemgarden.org/op/even_vs_odd_latin_squares.
Source subject path: Combinatorics.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/even_vs_odd_latin_square... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,042 | OPG-1797 | 2-accessibility of primes | Question Is the set of prime numbers 2-accessible? | Source: Open Problem Garden. Original node ID: 1797. URL: http://www.openproblemgarden.org/op/2_accessibility_of_primes.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/2_accessibility_of_pri... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: Oscar Quester proves that the primes have degree of accessibility exactly two, answering the Landman–Robertson question affirmatively. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof clai... | [
{
"url": "https://arxiv.org/abs/2606.00410",
"label": "Oscar Quester, The Primes are 2-Accessible, arXiv:2606.00410 (3 June 2026)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups... | 2026-09-27T00:00:00 | Oscar Quester, The Primes are 2-Accessible, arXiv:2606.00410 (3 June 2026).: https://arxiv.org/abs/2606.00410 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,043 | OPG-1825 | 3-accessibility of Fibonacci numbers | Question Is the set of Fibonacci numbers 3-accessible? | Source: Open Problem Garden. Original node ID: 1825. URL: http://www.openproblemgarden.org/op/3_accessibility_of_fibonacci_numbers.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/3_accessibi... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://arxiv.org/abs/2609.33744",
"checked_at": "2026-10-08",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] **solved (no; preprint)**. Landman and Robertson's question whether the Fibonacci numbers are 3-accessible. W. J. Wesley ([arXiv:2609.33744](https://arxiv.org/abs/2609.33744)) colours n by the class of ⌊αn⌋ mod 232, α = 184 + 8φ, under an explicit partition of the residues ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,044 | OPG-2063 | Wide partition conjecture | Conjecture An integer partition is wide if and only if it is Latin. | Source: Open Problem Garden. Original node ID: 2063. URL: http://www.openproblemgarden.org/op/wide_partition_conjecture.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/wide_partition_conject... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,045 | OPG-37167 | Shuffle-Exchange Conjecture | Given integers $k,n\ge2$, let $d(k,n)$ be the smallest integer $d\ge2$ such that the symmetric group $\frak S$ on the set of all words of length $n$ over a $k$-letter alphabet can be generated as $\frak S = (\sigma \frak G)^d:=\sigma\frak G \sigma\frak G \dots \sigma\frak G$ ( $d$ times), where $\sigma\in \frak S$ is t... | Source: Open Problem Garden. Original node ID: 37167. URL: http://www.openproblemgarden.org/op/shuffle_exchange_conjecture.
Source subject path: Combinatorics.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/shuffle_exchange_con... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | [
"OPG-37089"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: The shuffle-exchange formula d(k,n)=2n-1 is false: a 2026 preprint proves d(k,3)=6 for every k>=3, whereas the conjecture predicts 5. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof claim... | [
{
"url": "https://arxiv.org/abs/2607.15296",
"label": "P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and foll... | 2026-09-27T00:00:00 | P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026).: https://arxiv.org/abs/2607.15296 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,046 | OPG-37181 | Beneš Conjecture | Let $E$ be a non-empty finite set. Given a partition $\bf h$ of $E$, the stabilizer of $\bf h$, denoted $S(\bf h)$, is the group formed by all permutations of $E$ preserving each block of $\mathbf h$.
Problem ( $\star$ ) Find a sufficient condition for a sequence of partitions ${\bf h}_1, \dots, {\bf h}_\ell$ of $E$ t... | Source: Open Problem Garden. Original node ID: 37181. URL: http://www.openproblemgarden.org/op/bene_conjecture.
Source subject path: Combinatorics.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/bene_conjecture
- Author(s): Ben... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
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"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
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"created_at": "2026-07-31T15:26:25.671Z"
} | [
"OPG-37210"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: The partition-stabilizer Beneš conjecture in the source statement is refuted by explicit two-stage graph counterexamples to its stated inequality/form. This is a consistency correction based on that cited review; the discussion does not independently re-veri... | [
{
"url": "https://arxiv.org/abs/2607.15296",
"label": "P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and foll... | 2026-09-27T00:00:00 | P. Chojecki, Beneš and Shuffle-Exchange Counterexamples, arXiv:2607.15296 (2026).: https://arxiv.org/abs/2607.15296 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,047 | OPG-37222 | Dividing up the unrestricted partitions | Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the... | Source: Open Problem Garden. Original node ID: 37222. URL: http://www.openproblemgarden.org/op/dividing_up_the_unrestricted_partitions.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/dividin... | 1 | open | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,048 | OPG-37226 | Sequence defined on multisets | Conjecture Define a $2 \times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, ... | Source: Open Problem Garden. Original node ID: 37226. URL: http://www.openproblemgarden.org/op/a_sequence_defined_on_multisets.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/a_sequence_defi... | 1 | solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://doi.org/10.1080/00029890.1994.11996991",
"checked_at": "2026-10-08",
"kind": "resolution",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] **solved (yes)**. These are the counting sequences of V. Bronstein and A. S. Fraenkel ([Amer. Math. Monthly 101 (1994)](https://doi.org/10.1080/00029890.1994.11996991)): ultimately periodic; J. Sauerberg and L. Shu ([Amer. Math. Monthly 104 (1997)](https://doi.org/10.1080/0... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,049 | OPG-37228 | Square achievement game on an n x n grid | Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland ... | Source: Open Problem Garden. Original node ID: 37228. URL: http://www.openproblemgarden.org/op/a_game_on_an_n_x_n_grid.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/a_game_on_an_n_x_n_grid... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,050 | OPG-37230 | Transversal achievement game on a square grid | Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if any) to occupy a set of $n$ cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play? | Source: Open Problem Garden. Original node ID: 37230. URL: http://www.openproblemgarden.org/op/a_transversal_achievement_game_on_a_square_grid.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] . The note records a draw for every n > 1, from two preprints that their authors have since marked as wrong in the arXiv comments ([arXiv:2101.00770](https://arxiv.org/abs/2101.00770), [arXiv:2101.05065](https://arxiv.org/abs/2101.05065)). **The first player wins for every ... | null | Reconciled with the existing literature review of 2026-08-17: The transversal achievement game is a draw under optimal play for every n>1. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof claims.
**What remains.** The source question is answered; fu... | [
{
"url": "https://arxiv.org/abs/2101.00770",
"label": "N. Krishna, Transversal achievement game on a square grid, arXiv:2101.00770 (2021)."
},
{
"url": "https://arxiv.org/abs/2101.05065",
"label": "A. Dumitrescu, On a two-player transversal game on a square grid, arXiv:2101.05065 (2021)."
},
... | 2026-09-27T00:00:00 | N. Krishna, Transversal achievement game on a square grid, arXiv:2101.00770 (2021).: https://arxiv.org/abs/2101.00770
A. Dumitrescu, On a two-player transversal game on a square grid, arXiv:2101.05065 (2021).: https://arxiv.org/abs/2101.05065 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,051 | OPG-37416 | Length of surreal product | Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$, where $o$ is some ordinal. This $o$ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $s$ as $\ell(s)$.
It is easy to prove that
$$\ell(s+t... | Source: Open Problem Garden. Original node ID: 37416. URL: http://www.openproblemgarden.org/op/length_of_surreal_product.
Source subject path: Combinatorics.
Source importance: Low ✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/length_of_surreal_product... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,052 | OPG-58213 | Roller Coaster permutations | Let $S_n$ denote the set of all permutations of $[n]=\set{1,2,\ldots,n}$. Let $i(\pi)$ and $d(\pi)$ denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in $\pi$. Let $X(\pi)$ denote the set of subsequences of $\pi$ with length at least three. Let $t(\pi)$ denote $\s... | Source: Open Problem Garden. Original node ID: 58213. URL: http://www.openproblemgarden.org/op/roller_coaster_permutations.
Source subject path: Combinatorics.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/roller_coaster_permu... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,053 | OPG-60000 | The Double Cap Conjecture | Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\pi/4$ around the north and south poles. | Source: Open Problem Garden. Original node ID: 60000. URL: http://www.openproblemgarden.org/op/the_double_cap_conjecture.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/the_double_cap_conjec... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,054 | OPG-60002 | Saturation in the Hypercube | Question What is the saturation number of cycles of length $2\ell$ in the $d$-dimensional hypercube? | Source: Open Problem Garden. Original node ID: 60002. URL: http://www.openproblemgarden.org/op/saturation_in_the_hypercube.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/saturation_in_the_h... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,055 | OPG-60003 | Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube | Problem Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube. | Source: Open Problem Garden. Original node ID: 60003. URL: http://www.openproblemgarden.org/op/extremal_4_neighbour_bootstrap_percolation_in_the_hypercube.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemg... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,056 | OPG-60006 | Turán Problem for $10$-Cycles in the Hypercube | Problem Bound the extremal number of $C_{10}$ in the hypercube. | Source: Open Problem Garden. Original node ID: 60006. URL: http://www.openproblemgarden.org/op/turan_problem_for_10_cycles_in_the_hypercube.
Source subject path: Combinatorics.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/tu... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,057 | OPG-37196 | Perfect 2-error-correcting codes over arbitrary finite alphabets. | Conjecture Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code? | Source: Open Problem Garden. Original node ID: 37196. URL: http://www.openproblemgarden.org/op/perfect_2_error_correcting_codes_over_arbitrary_finite_alphabets.
Source subject path: Combinatorics > Codes.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,058 | OPG-762 | Combinatorial covering designs | A $(v, k, t)$ covering design, or covering, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by $C(v, k, t)$.
Problem Find a closed form... | Source: Open Problem Garden. Original node ID: 762. URL: http://www.openproblemgarden.org/op/covering_designs.
Source subject path: Combinatorics > Designs.
Source importance: Low ✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/covering_designs
- Author(... | 1 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,059 | OPG-148 | A nowhere-zero point in a linear mapping | Conjecture If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, then there are column vectors $x,y \in {\mathbb F}^n$ which have no coordinates equal to zero such that $Ax=y$. | Source: Open Problem Garden. Original node ID: 148. URL: http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping.
Source subject path: Combinatorics > Matrices.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/o... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,060 | OPG-150 | The additive basis conjecture | Conjecture For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$ ) so that the union (as multisets) of any $c(p)$ bases of the vector space $({\mathbb Z}_p)^n$ contains an additive basis. | Source: Open Problem Garden. Original node ID: 150. URL: http://www.openproblemgarden.org/op/the_additive_basis_conjecture.
Source subject path: Combinatorics > Matrices.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/the_addit... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,061 | OPG-151 | The permanent conjecture | Conjecture If $A$ is an invertible $n \times n$ matrix, then there is an $n \times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero. | Source: Open Problem Garden. Original node ID: 151. URL: http://www.openproblemgarden.org/op/the_permanent_conjecture.
Source subject path: Combinatorics > Matrices.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/the_permanent... | 1 | open | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,062 | OPG-152 | The Alon-Tarsi basis conjecture | Conjecture If $B_1,B_2,\ldots B_p$ are invertible $n \times n$ matrices with entries in ${\mathbb Z}_p$ for a prime $p$, then there is a $n \times (p-1)n$ submatrix $A$ of $[B_1 B_2 \ldots B_p]$ so that $A$ is an AT-base. | Source: Open Problem Garden. Original node ID: 152. URL: http://www.openproblemgarden.org/op/the_alon_tarsi_basis_conjecture.
Source subject path: Combinatorics > Matrices.
Source importance: Medium ✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/the_al... | 1 | open | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
3,063 | OPG-361 | Rota's unimodal conjecture | Let $M$ be a matroid of rank $r$, and for $0 \le i \le r$ let $w_i$ be the number of closed sets of rank $i$.
Conjecture $w_0,w_1,\ldots,w_r$ is unimodal.
Conjecture $w_0,w_1,\ldots,w_r$ is log-concave. | Source: Open Problem Garden. Original node ID: 361. URL: http://www.openproblemgarden.org/op/rotas_unimodal_conjecture.
Source subject path: Combinatorics > Matroid Theory.
Source importance: High ✭✭✭.
Discussion and literature:
Source-derived literature/context:
- Source: http://www.openproblemgarden.org/op/rotas_u... | 2 | partially_solved | null | null | 2 | 12 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 12,
"name": "opengarden",
"display_name": "OpenGarden",
"description": "Problems imported from the Open Problem Garden collection, using source statements, discussion notes, and subject metadata.",
"slug": "opengarden",
"order_index": 12,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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