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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,864 | KP-3.66 | Kirby Problem 3.66 | Suppose that $Y$ is a closed, oriented 3-manifold, and let $S_{2,\infty}(Y)$ denote the Kauffman bracket skein module over $R=\mathbb{Z}[A,A^{-1}]$ as in Problem 3.65.
(a) Is the dimension
$$
\dim_{\mathbb{Q}(A)} S_{2,\infty}(Y)\otimes_R \mathbb{Q}(A)
$$
equal to the dimension of the degree 0 part of Abouzaid--Manol... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.66.
Literature notes:
(1) An affirmative answer to the first question was conjectured in [GJS23], which also contains the proof that
$$
S_{2,\infty}(Y)\otimes_R \mathbb{Q}(A)
$$
is finite dimensional over $\mathbb{Q}(q)$.
(2) ... | 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 |
2,865 | KP-3.67 | Kirby Problem 3.67 | Categorify the Witten--Reshetikhin--Turaev invariants of 3-manifolds. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.67.
Literature notes:
The Witten--Reshetikhin--Turaev (WRT) invariants [Wit89, RT91] are maps from the set of roots of unity to C. They are an analogue of the Jones polynomial for 3-manifolds. Khovanov [Kho00] constructed a categ... | 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 |
2,866 | KP-3.68 | Kirby Problem 3.68 | (a) Give a mathematical definition of the $\widehat{Z}$ invariants for all 3-manifolds.
(b) Categorify the $\widehat{Z}$ invariants. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.68.
Literature notes:
(1) In [GPV17, GPPV20], Gukov, Pei, Putrov, and Vafa conjectured the existence of some power series with integral coefficients associated to 3-manifolds, denoted $\widehat{Z}_a(q)$. They converge in the unit... | 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 |
2,867 | KP-3.69 | Kirby Problem 3.69 | (a) What is the isomorphism type of $\Theta^3_{\mathbb{Z}}$?
(b) Does there exist a torsion element $[Y]$ in $\Theta^3_{\mathbb{Z}}$?
(c) Does there exist a torsion element $[Y]\in\Theta^3_{\mathbb{Z}}$ having Rokhlin invariant $\mu([Y])=1$?
(d) Do there exist infinitely-divisible elements in $\Theta^3_{\mathbb{Z}}$... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.69.
Literature notes:
(1) The integer homology cobordism group $\Theta^3_{\mathbb{Z}}$ is known to be a countable abelian group that has a summand isomorphic to $\mathbb{Z}^{\infty}$ by work of [DHST23].
(2) In dimension $n=1,2$... | 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] , `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 |
2,868 | KP-3.70 | Kirby Problem 3.70 | Is $\Theta^3_{\mathbb{Z}}$ generated by the classes of knot surgeries $[S^3_{1/n}(K)]$, where $n$ ranges over all integers and $K$ ranges over all knots in $S^3$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.70.
Literature notes:
There are homology spheres that are not given by surgery on a knot in $S^3$ [GL89, Auc97, HKL16b]. Nozaki--Sato--Taniguchi used filtered instanton Floer homology to exhibit a homology 3-sphere which is not e... | 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**: surgeries on knots of bounded genus do not generate (Hendricks–Hom–Lidman) and some classes are not a single surgery (Nozaki–Sato–Taniguchi), which leaves generation by all knot surgeries undecided. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,869 | KP-3.71 | Kirby Problem 3.71 | Is there a nontrivial element in the kernel of the natural map
$$
\Theta^3_{\mathbb{Z}}\longrightarrow \Theta^3_{\mathbb{Z}/2\mathbb{Z}};
$$
that is, does there exist an integer homology 3-sphere that does not bound a smooth $\mathbb{Z}$-homology 4-ball but does bound a smooth $\mathbb{Z}/2\mathbb{Z}$-homology 4-ball... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.71.
Literature notes:
There is a natural homomorphism from the concordance group $C$ to $\Theta^3_{\mathbb{Z}/2\mathbb{Z}}$ via taking double branched covers. As discussed in Problem 3.72, $\Sigma(2,3,7)$ bounds a rational homolo... | 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 |
2,870 | KP-3.72 | Kirby Problem 3.72 | (a) Does the kernel of the map $\Theta^3_{\mathbb{Z}}\to\Theta^3_{\mathbb{Q}}$ contain a subgroup that is isomorphic to $\mathbb{Z}^{\infty}$?
(b) If so, does it contain such a subgroup as a summand? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.72.
Literature notes:
Fintushel and Stern [FS84] showed that $\Sigma(2,3,7)=S^3_{+1}(4_1)$ bounds a smooth rational homology ball; as it has nontrivial Rokhlin invariant, it is a nontrivial element of the kernel of the map above.... | 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 |
2,871 | KP-3.73 | Kirby Problem 3.73 | (a) Calculate $\Theta^{\mathrm{TOP}}_{\mathbb{Z}/p}$.
(b) Calculate $\Theta^{\mathrm{TOP}}_{\mathbb{Q}}$.
(c) Is the linking form homomorphism $[\operatorname{lk}]$ injective? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.73.
Literature notes:
(1) Freedman proved that any integral homology 3-sphere bounds a contractible topological 4-manifold, unique up to homeomorphism rel boundary [FQ90, Corollary 9.3C], [Fre82, Theorem 1.41][BKK+21, Chapter 21.... | 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 |
2,872 | KP-3.74 | Kirby Problem 3.74 | Let $Y$ be a rational homology sphere and $f:Y\to Y$ be a self-diffeomorphism of $Y$. Suppose $W$ is a 4-manifold with boundary $Y$ such that $f$ extends to a self-diffeomorphism of $W$.
(a) What constraints are there on the intersection form of $W$?
(b) Does there exist a pair $(Y,f)$ such that $Y$ bounds a homology... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.74.
Literature notes:
(1) It is known that the 3-dimensional bordism-with-diffeomorphism group is trivial [Mel79]; that is, if $Y$ is a 3-manifold equipped with a self-diffeomorphism $f$, then there always exists a smooth 4-manif... | 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 |
2,873 | KP-3.75 | Kirby Problem 3.75 | Let $Y$ be a rational homology 3-sphere equipped with an action of a cyclic group $\mathbb{Z}/p\mathbb{Z}$. Suppose $W$ is a 4-manifold with boundary $Y$ over which the action of $\mathbb{Z}/p\mathbb{Z}$ extends smoothly.
(a) What constraints are there on the intersection form of $W$?
(b) Does there exist an integer ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.75.
Literature notes:
While this is similar to Problem 3.74, the motivation and techniques for the present question are slightly different. If $f:Y\to Y$ generates the action of a cyclic group, then obstructing the extension 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 |
2,874 | KP-3.76 | Kirby Problem 3.76 | What is the structure of the equivariant homology cobordism groups? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.76.
Literature notes:
There are several equivariant homology cobordism groups one can construct. One version is the $\mathbb{Z}/p\mathbb{Z}$-equivariant cobordism group as appropriate
homology cobordism classes of pairs consisti... | 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 |
2,875 | KP-3.77 | Kirby Problem 3.77 | Does there exist a hyperbolic rational homology 3-sphere that is the totally geodesic boundary of a compact, orientable hyperbolic 4-manifold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.77.
Literature notes:
(1) When a compact hyperbolic 3-manifold is the totally geodesic boundary of a compact hyperbolic 4-manifold, it is said to geometrically bound. Questions about geometric boundaries of hyperbolic manifolds o... | 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 rational homology spheres of Ferrari–Kolpakov–Reid bound only nonorientable hyperbolic 4-manifolds, and the question asks for an orientable one. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,876 | KP-3.78 | Kirby Problem 3.78 | (a) Is there a non-semisimple 3-TQFT whose mapping class group representation is faithful or has an element in its kernel?
(b) Define a 4-manifold invariant via non-semsimple categories that can distinguish 4-manifolds that are not distinguished by classical or gauge-theoretic invariants.
(c) Is there is a full 3-TQF... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.78.
Literature notes:
(1) In 1989, Atiyah [Ati88b] gave the definition of an n-dimensional topological quantum field theory (n-TQFT for short) that assigns objects in some category to n-manifolds and morphisms to cobordisms. Many... | 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 |
2,877 | KP-4.1 | Kirby Problem 4.1 | (4-dimensional Poincaré conjecture). Is there a unique smooth structure on the 4-sphere? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.1.
Literature notes:
(1) This appears as Problems 4.45 and 4.89 in [Kir97].
(2) This is one of the remaining open cases of the smooth Poincaré Conjecture. In dimensions at most three, uniqueness of smooth structures follows from... | 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 |
2,878 | KP-4.2 | Kirby Problem 4.2 | Does every smooth, closed 4-manifold admit an exotic smooth structure? Infinitely many? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.2.
Literature notes:
(1) The question is open for many families of 4-manifolds. See also Problems 4.3 and 4.4.
(2) A version of this problem in [Kir97, Problem 4.86] raises the same question for complex algebraic surfaces. Evide... | 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 |
2,879 | KP-4.3 | Kirby Problem 4.3 | Are there exotic smooth structures on the following closed, simply-connected 4–manifolds?
(a) $\#_{k}\mathbb{CP}^{2}$ for any $k \geq$ 1.
(b) $\#_{m}(\mathbb{CP}^{2}\#\overline{\mathbb{CP}}{}^{2})$ and $\#_{n}(S^{2} \times S^{2})$ for $m \leq$ 8, $n \leq$ 10.
(c) $\#_{p}K3 \#_{q}(S^{2} \times S^{2})$ for any $p \geq... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.3.
Literature notes:
(1) These are the main families of closed, simply connected 4–manifolds, besides $S^{4}$, not known to admit exotic smooth structures. Namely, those that are (a) definite, (b) $\sigma =$ 0 with $e <$ 20, 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 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,880 | KP-4.4 | Kirby Problem 4.4 | Is there an exotic smooth structure on some product 4-manifold $S^{1} \times Y^{3}$ or $\Sigma_{g} \times \Sigma_{h}$? Do they all admit exotic smooth structures? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.4.
Literature notes:
(1) A homeomorphism classification is unavailable for many of these 4–manifolds, making it challenging to determine whether a potential exotic copy falls within the desired homeomorphism class.
(2) Significa... | 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 |
2,881 | KP-4.5 | Kirby Problem 4.5 | Does every connected, open 4-manifold admit uncountably many smooth structures? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.5.
Literature notes:
(1) Recall that a manifold is said to be open if it has empty boundary and every component is noncompact. Any such 4-manifold is smoothable by work of Quinn [Qui82, Corollary 2.2.3] (see also [FQ90, Theorem 8... | 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 |
2,882 | KP-4.6 | Kirby Problem 4.6 | Does every closed, orientable 3-manifold bound an absolutely exotic pair of smooth, orientable 4-manifolds? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.6.
Literature notes:
(1) Yasui [Yas12], Etnyre–Min–Mukherjee [EMM22], and Iida-MukherjeeTaniguchi [IMT25] gave sufficient conditions on a closed, connected, orientable 3-manifold Y to bound a compact 4-manifold X with $\partial X... | 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 |
2,883 | KP-4.7 | Kirby Problem 4.7 | (a) If $M_{1},M_{2}are$ two homeomorphic closed, oriented 4-manifolds, is $M_{1}\#S^{2} \times S^{2}$ diffeomorphic to $M_{2}\#S^{2} \times S^{2}$?
(b) Is there a fixedn>0such that $M_{1}\#_{n}S^{2} \times S^{2}$ is diffeomorphic to $M_{2}\#_{n}S^{2} \times S^{2}$ for every pair $M_{1}, M_{2}$ as above? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.7.
Literature notes:
(1) By Gompf’s generalization of Wall’s theorem [Wal64b, Gom84], for any given pair $M_{1}, M_{2}$ as above, there exists $k \geq$ 0 such that $M_{1}\#_{k}S^{2} \times S^{2}$ is diffeomorphic to $M_{2}\#_{k}S... | 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 |
2,884 | KP-4.8 | Kirby Problem 4.8 | Let X be a closed, simply connected, smooth 4-manifold, and T a smoothly embedded torus in X with $\pi_{1}(X$ −T) =1 and $[T]^{2}$ =0. Let $X_{K}$ be the result of Fintushel–Stern knot surgery on X along a knot K. If $K_{1}$ and $K_{2}$ are two prime knots such that $X_{K,1}$ and $X_{K,2}$ are diffeomorphic, does it fo... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.8.
Literature notes:
(1) Fintushel and Stern showed that $X_{K}$ is homeomorphic to X, and $SW(X_{K}) = SW(X) \cdot\Delta_{K}$, where SW denotes the Seiberg-Witten series and $\Delta_{K}$ the Alexander polynomial.
(2) One can as... | 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 |
2,885 | KP-4.9 | Kirby Problem 4.9 | Is every Gluck twist in $S^{4}$ standard? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.9.
Literature notes:
(1) This is [Kir78, Problem 4.24]. Given an embedded 2-sphere in $S^{4}$, Gluck defined a surgery operation that yields a homotopy 4-sphere [Glu62], which is in fact homeomorphic to $S^{4}$ due to Freedman [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 |
2,886 | KP-4.10 | Kirby Problem 4.10 | (a) Is every homotopy $B^{4}$ with boundary $S^{3}$ obtained by performing a Gluck twist on some knotted 2-sphere in $B^{4}$?
(b) Suppose a homotopy 4-ball X is obtained by performing a Gluck twist on a knotted 2-sphere in $B^{4}$. Is $X \times I$ necessarily diffeomorphic to $B^{5}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.10.
Literature notes:
(1) Habiro, Marumoto and Yamada [HMY00] showed that if a homotopy 4-ball is obtained by performing Gluck twists on multiple 2-spheres in $B^{4}$, then it can be obtained by a Gluck construction on a single k... | 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/10.1016/j.aim.2025.110455",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**; the point the note leaves for checking is fine. K3 (2026), Problem 4.10 (pp. 198–199, proposed by D. Gabai) asks (a) whether every homotopy 4-ball is a Gluck twist on a 2-sphere in B⁴ and (b) whether X × I ≅ B⁵ for every such Gluck twist X. Gabai–... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,887 | KP-4.11 | Kirby Problem 4.11 | Let M be a smooth 4-manifold and letf: $S^{2} \to M$ be a smooth embedding with trivial normal bundle. Then let $M_{f}$ denote the result of Gluck twisting on M along f.
(a) Does there exist an orientable M, and an embedding f, such that M and $M_{f}$ are homeomorphic but not diffeomorphic?
(b) Does there exist an or... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.11.
Literature notes:
(1) The version of this question with $M =S^{4}$ has received considerable interest (see Problems 4.9 and4.10). It is possible that the general case might be attacked more readily and/or provide insight to 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 | {
"source_url": null,
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **open**: the only Gluck twists known to change the smooth structure are in nonorientable manifolds (Akbulut), and both parts ask for an orientable one. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,888 | KP-4.12 | Kirby Problem 4.12 | For X a closed simply connected smooth 4-manifold, let $g_{X}: H_{2}(X) \to \mathbb{N}$ denote the genus function, which assigns to every homology class the minimal genus of a smooth embedded surface representing that class.
(a) Suppose $f: X_{1} \to X_{2}$ is a homeomorphism between simply connected, smooth, closed 4... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.12.
Literature notes:
(1) A version of this problem appeared in Stipsicz–Szabó [SS24b].
(2) There are several ways one might define a genus function for a 4-manifold with boundary. For concreteness, we present one. For X a simpl... | 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/2501.18584",
"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. K3 (2026), Problem 4.12 (pp. 199–200, scribed by L. Piccirillo) asks whether the genus function determines the diffeomorphism type (a) of closed simply connected 4-manifolds and (b), rel boundary and with relative genus... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,889 | KP-4.13 | Kirby Problem 4.13 | (a) Does every large $\mathbb{R}^{4}-homeomorph$ lie $in\mathcal{R}_{K}$ for some Kthat is not smoothly slice?
(b) Does there exist an infinite sequence of knots ${K_{i}}$ such that $\mathcal{R}_{K,i} \ne \mathcal{R}_{K,j}$ whenever $i \ne j$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.13.
Literature notes:
(1) Given a knot K, let $X_{K}$ denote its 0-trace, which is obtained, by definition, by attaching a 0-framed 2-handle to $B^{4}$ along K and smoothing corners. Reserve the symbol $\mathbb{R}^{4}$ to refer 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 | {
"source_url": null,
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **open**: only three knots with pairwise distinct sets R_K are known, and infinitely many only for links (Gompf); nothing is known towards (a). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,890 | KP-4.14 | Kirby Problem 4.14 | Is there a universal cork? More precisely, does there exist some cork (C, f) such that given any pair W and $W^{1}$ of closed, simply connected 4-manifolds that are homeomorphic but not diffeomorphic, there exists a smooth embedding $C \hookrightarrow W$ such that (W $\setminus$ Int C) $\cup _{f} C$ is diffeomorphic to... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.14.
Literature notes:
(1) A cork (C, f) is a pair of a compact, contractible, smooth 4-manifold C and a diffeomorphism $f: \partial C \to \partial C$, such that f does not extend to a diffeomorphism of C. Some authors require f t... | 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 |
2,891 | KP-4.15 | Kirby Problem 4.15 | (11/8 Conjecture). Does every smooth, spin, closed 4-manifold X satisfy $b_{2}(X) \geq 11|\sigma(X)|$, where $\sigma(X)$ is the signature of the intersection form? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.15.
Literature notes:
(1) This is [Kir97, Problem 4.92], originally raised by Matsumoto.
(2) A positive resolution would answer the Geography Question 4.54 for the smooth simply connected case. By Rokhlin’s theorem, the intersec... | 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 the statement**: the fraction is lost. K3 (2026), Problem 4.15 (p. 202, scribed by H. Konno and C. Manolescu; Problem 4.92 of Kirby's 1997 list, raised by Matsumoto) is the 11/8 conjecture: does every smooth closed spin 4-manifold X sati... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,892 | KP-4.16 | Kirby Problem 4.16 | (a) Do there exist closed, oriented, smooth, irreducible 4-manifolds with $b^{+}_{2} >$ 1 and $c^{2}_{1}:=2\chi+3\sigma<0$?
(b) Is there an irreducible exotic smooth structure on $\mathbb{CP}^{2}\#_{n}\mathbb{CP}^{2}$ for n>9? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.16.
Literature notes:
(1) $S^{2}–bundles$ over $\Sigma_{h}$ are irreducible 4–manifolds with $b^{+}_{2} =$ 1, and for $h \geq$ 2, they have $c^{2}_{1}<0$.
(2) An exotic $\mathbb{CP}^{2}\#_{n}\mathbb{CP}^{2}$ for $n >$ 9 would ha... | 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": null,
"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 (b)**: the overline is lost. K3 (2026), Problem 4.16 (p. 203, proposed by R. Fintushel and R. Stern) asks (a) for closed oriented smooth irreducible 4-manifolds with b₂⁺ > 1 and c₁² := 2χ + 3σ < 0, and (b) for an irreducible exotic ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,893 | KP-4.17 | Kirby Problem 4.17 | Is there an irreducible, closed, simply connected, oriented 4– manifold with $b^{+}_{2}$ and $b^{-}_{2}$ both even? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.17.
Literature notes:
(1) The condition $b^{+}_{2}(X)$ and $b^{-}_{2}(X)$ both being even is equivalent to the holomorphic Euler characteristic $\chi_{h}(X):=\frac{1}{4}(e(X)+\sigma(X))$ not being an integer for either orientatio... | 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 |
2,894 | KP-4.18 | Kirby Problem 4.18 | (a) Does there exist a pair of smooth, closed 4-manifolds that are homotopy equivalent but not simple homotopy equivalent?
(b) Does there exist a pair of smoothlyh-cobordant, smooth, closed 4-manifolds that are not smoothly s-cobordant? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.18.
Literature notes:
(1) There exist examples of topologicallyh-cobordant 4-manifolds that are not topologically s-cobordant [KPR22], and examples of topological manifolds that are homotopy equivalent but not simple homotopy equ... | 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/2604.27635",
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **open**: the known examples (Kasprowski–Powell–Ray; Nagy–Nicholson–Powell) are topological 4-manifolds, and both questions ask for smooth ones; Kupers–Powell, [arXiv:2604.27635](https://arxiv.org/abs/2604.27635), only exclude smooth examples for (b) for some fundamental gr... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,895 | KP-4.19 | Kirby Problem 4.19 | What are the possible Euler characteristics of closed, aspherical 4-manifolds? More specifically, we ask the following.
(a) Is it always the case that $\chi \geq |\sigma|$?
(b) What is the smallest Euler characteristic of a closed hyperbolic 4-manifold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.19.
Literature notes:
(1) The questions are motivated by the following conjecture.
\paragraph{Conjecture.} The Euler characteristic of X is non-negative. This is [Kir97, Problem 4.10]. One can ask the question for all aspherical... | 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 |
2,896 | KP-4.20 | Kirby Problem 4.20 | Is $*\mathbb{RP}^{4}\#*\mathbb{RP}^{4}$ smoothable? Is *En\#*En smoothable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.20.
Literature notes:
(1) The manifold $*\mathbb{RP}^{4}$ was first constructed by Ruberman [Rub84] (see also [HKT94]). It is homotopy equivalent to $\mathbb{RP}^{4}$ but not homeomorphic to $\mathbb{RP}^{4}$. In particular, it h... | 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 |
2,897 | KP-4.21 | Kirby Problem 4.21 | Is every topological closed 4–manifold M the union of submanifolds $Y \cup Z$, where Y is smoothable, Z is acyclic, and $Y \cap Z$ is their common boundary, a homology sphere $\Sigma$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.21.
Literature notes:
(1) This is [Kir97, Problem 4.74].
(2) In the case that M is simply connected the result holds. Freedman [Fre82, Theorem 1.5] proved that the homeomorphism type of a simply connected closed 4–manifold is de... | 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 |
2,898 | KP-4.22 | Kirby Problem 4.22 | Let $\pi$ be a good group, and let X be a smooth 4-manifold with $\pi_{1}(X) = \pi$. Does $L^{s}_{5}(\mathbb{Z}[\pi])$ act on the smooth structure set of X, in such a way that the action reduces under the forgetful map to the Wall realization action on the topological structure set? In particular, can this be done when... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.22.
Literature notes:
(1) The surgery exact sequence, valid for higher dimensional manifolds, ends with $\mathcal{N}(X \times I,\partial(X \times$ I)) $\to L^{s}_{n,1}(\mathbb{Z}\pi) \to ^{\Theta} \to \mathcal{S}(X) \to \mathcal{... | 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 |
2,899 | KP-4.23 | Kirby Problem 4.23 | (Schoenflies problem). If $\Sigma$ is a smoothly embedded $S^{3}$ in $S^{4}$, then its closed complements are smooth 4-balls. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.23.
Literature notes:
(1) This is the famous Schoenflies problem, which appears in [Kir97, Problem 4.32].
(2) Progress on this old and important conjecture has, sadly, been more aspirational than concrete. Scharlemann proved thi... | 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 |
2,900 | KP-4.24 | Kirby Problem 4.24 | Let K be a framed knot in $S^{3} = \partial B^{4}$. Let U be a meridian of K. Does there exist a smoothly embedded disk D in $B^{4} \cup _{\nu K} h^{2}$ bounded by U that intersects the cocore of $h^{2}$ algebraically zero times, such that the 4-manifold obtained by removing an open tubular neighborhood of the disk is ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.24.
Literature notes:
(1) We represent the situation of this problem in Figure 1. Note the “big dot” placed on U. An “ordinary dot” on an unknot is an instruction to carve the standard disk (in $B^{4})$ bounded by the unknot. An ... | 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 |
2,901 | KP-4.25 | Kirby Problem 4.25 | Under what conditions does a closed, orientable 3-manifold M smoothly embed in $S^{4}$? Is this question algorithmically decidable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.25.
Literature notes:
(1) This is in [Kir78, Problem 3.20]. As noted there, every homology 3sphere embeds topologically in $S^{4}$ [Fre82]. Smoothly, even for homology spheres, there is an obstruction coming from the Rokhlin inva... | 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 |
2,902 | KP-4.26 | Kirby Problem 4.26 | If Y is a homology three-sphere, does the punctured manifold $Y_{0}$ =Y $\setminus \operatorname{Int}(B^{3})$ smoothly embed in $S^{4}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.26.
Literature notes:
(1) Every homology 3-sphere embeds topologically in $S^{4}$ (see [FQ90, Corollary 9.3C]). The Poincaré homology sphere does not embed smoothly in $S^{4}$, since it has nonzero Rokhlin invariant. Every homolo... | 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 |
2,903 | KP-4.27 | Kirby Problem 4.27 | Find exotic 3-balls in $S^{4}$, considered up to isotopy rel. boundary. That is, find a pair of 3-balls $B_{1}, B_{2}$ smoothly embedded in $S^{4}$ with the same boundary such that $B_{1}, B_{2}$ are topologically but not smoothly isotopic rel. boundary. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.27.
Literature notes:
(1) In [BG19, Section 9], Budney–Gabai observe that if no such 3-balls exist, then the smooth 4-dimensional Schoenflies Conjecture (Problem 4.23) is true.
(2) Budney–Gabai [BG19] find infinitely many 3-ball... | 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 knotted 3-balls of Budney–Gabai are not even topologically isotopic rel boundary; no pair that is topologically but not smoothly isotopic is known. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,904 | KP-4.28 | Kirby Problem 4.28 | Every closed, orientable 3-manifold embeds smoothly in some connected sum of copies of $S^{2} \times S^{2}$. Given a closed 3-manifold M, let $s(M) \geq$ 0 denote the smallest integer such that M embeds smoothly (respectively locally flatly) in a connected sum of $s(M)$ copies of $S^{2} \times S^{2}$. Let $s(M) \geq$ 0... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.28.
Literature notes:
Manifolds with $s(M)$ =0 are the topic of Problem 4.25. The functionswas introduced and studied by [AGL17]. That paper includes a comprehensive summary of the literature. Note in particular $thats(M)$ =1 imp... | 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/2607.27619",
"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: part (b) is answered (yes, by an unrefereed note), and **fix the statement**. K3 (2026), Problem 4.28 (pp. 212–213, scribed by M. Powell) lets s(M), resp. s̃(M), be the least number of copies of S² × S², resp. of the tw... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,905 | KP-4.29 | Kirby Problem 4.29 | Let $\Sigma$ be a locally flat surface in $S^{4}$ with $\pi_{1}(S^{4} \setminus \Sigma)$ cyclic.
(a) Prove that $\Sigma$ is topologically unknotted.
(b) Assuming that $\Sigma$ is smoothly embedded and orientable, prove that $\Sigma$ is smoothly unknotted (or find an example that is not). | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.29.
Literature notes:
(1) The surface $\Sigma$ is known to be topologically unknotted in many cases.
\noindent$\bullet$ If $\Sigma=S^{2}$, the claim holds by work of Freedman–Quinn (see [FQ90]).
\noindent$\bullet$ If $\Sigma$ i... | 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/2603.22202",
"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: part (a) is now false in general, claimed for all orientable surfaces, and open only for nonorientable genus 6, 7, 8. K3 (2026), Problem 4.29 (pp. 213–214, scribed by M. Miller) asks to prove that a locally flat surface... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,906 | KP-4.30 | Kirby Problem 4.30 | Does there exist a pair of closed, oriented surfaces in $S^{4}$ that are topologically but not smoothly isotopic? If such an exotic pair exists, does there exist an infinite family? Can the surfaces be taken to be 2-spheres? Surfaces of genus-g for any given $g \geq$ 0? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.30.
Literature notes:
(1) There are many examples of exotic oriented surfaces in general 4-manifolds. These include infinite families of pairwise exotic 2-spheres (existence from [Wal64a], with smooth obstruction from work of Don... | 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 |
2,907 | KP-4.31 | Kirby Problem 4.31 | Does every knot in $S^{3}$ bound an exotic pair of orientable surfaces in $B^{4}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.31.
Literature notes:
(1) By an exotic pair, we mean two surfaces with the same knot as boundary, that are topologically but not smoothly isotopic rel. boundary.
(2) It is known that certain knots bound exotic genus one surfaces... | 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] `10.2140/gt.2021.25.2963`, cited as "Knot cobordisms, bridge index, and torsion in Floer homology": the DOI is "Transverse invariants and exotic surfaces in the 4-ball" by the same authors; the cited title is J. Topol. 13 (2020), 1701–1724, `10.1112/topo.12170`. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,908 | KP-4.32 | Kirby Problem 4.32 | Does there exist a locally flat embedding $f: \Sigma \to S^{4}$ for some closed surface $\Sigma$ such that f is not topologically ambiently isotopic to a smooth embedding? Particularly in the case that $\Sigma=S^{2}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.32.
Literature notes:
(1) There are many examples of this behavior in other 4-manifolds. In particular, suppose there is a locally flat embedding of a genus-g surface into a smooth 4-manifold X such that there is no smooth genus-... | 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/2603.22202",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **partially_solved (existence: yes, by an unrefereed preprint; the S² case is open)**. K3 (2026), Problem 4.32 (p. 216, proposed by A. Ray) asks for a locally flat embedding of a closed surface in S⁴ that is not topologically ambiently isotopic to a smooth embedding, partic... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,909 | KP-4.33 | Kirby Problem 4.33 | Let $S_{1}, S_{2}$ be two topologically isotopic, smoothly embedded closed surfaces in a closed, oriented, smooth 4-manifold X. When do $S_{1}$ and $S_{2}$ in X become smoothly isotopic after one
(a) external stabilization?
(b) standard internal stabilization?
(c) internal stabilizations? Is there an upper bound dep... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.33.
Literature notes:
(1) We say that (X, $S_{i})\#(S^{2} \times S^{2},\emptyset)$ is an external stabilization of (X, $S_{i})$, that is, we enlarge the topology of the surface complement by taking a connected sum with $S^{2} \ti... | 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] [arXiv:2405.07928](https://arxiv.org/abs/2405.07928): Daniel A. P. Galvin, "The Casson-Sullivan invariant for homeomorphisms of 4-manifolds" (cited: David Galvin, "Stable isotopy of surfaces in simply connected 4-manifolds"). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,910 | KP-4.34 | Kirby Problem 4.34 | Let $\Sigma$ be a surface embedded in $S^{4}$. Can $\Sigma$ be unknotted by a sequence of torus surgeries in its complement, such that the ambient manifold is always $S^{4}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.34.
Literature notes:
(1) This problem may be addressed in the smooth or topological locally flat categories. This would be an analogue of the 3-dimensional fact that any knot in $S^{3}$ can be transformed into the unknot via $\p... | 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 |
2,911 | KP-4.35 | Kirby Problem 4.35 | (a) Give an algebraic classification of groups that arise as the fundamental group of the complement of a smooth or locally flat 2-sphere in $S^{4}$.
(b) Let $\Lambda = \mathbb{Z}[t, t-^{1}]$. Classify the $\Lambda-modules$ that arise as the Alexander module of a smooth or locally flat 2-sphere in $S^{4}$. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.35.
Literature notes:
(1) This is [Kir97, Problem 1.48]. Hillman discusses this problem in the 2022 update of [Hil02, Part III, §14.6–14.10]. The analogous problem is also open for classical knots in $S^{3}$.
(2) Kervaire [Ker65... | 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 |
2,912 | KP-4.36 | Kirby Problem 4.36 | Are homotopy types of 2-knot complements determined by their homotopy 2-types? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.36.
Literature notes:
(1) The homotopy 2-type of a topological space X is a triple $(\pi_{1}(X), \pi_{2}(X)$, k(X)), where $k(X) \in H^{3}(\pi_{1}(X);\pi_{2}(X))$ is the first k-invariant of X [MW50]. Martins [Mar09] gave an algo... | 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 |
2,913 | KP-4.37 | Kirby Problem 4.37 | (Kinoshita conjecture). Does every projective plane in $S^{4}$ decompose as a connected sum of a knotted 2-sphere and an unknotted projective plane? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.37.
Literature notes:
(1) This question can be asked in either the smooth or the locally flat categories.
(2) The conjecture that the answer to this question is “yes,” is known as the Kinoshita Conjecture. This is named after th... | 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: Kinoshita's conjecture was disproved in May 2026 by a smooth projective plane in S4 that is not even topologically a connected sum of a 2-knot with an unknotted projective plane. This is a consistency correction based on that cited review; the discussion doe... | [
{
"url": "https://arxiv.org/abs/2605.12921",
"label": "Mark Hughes, Seungwon Kim, Maggie Miller, and Gheehyun Nahm, An irreducible real projective plane in the 4-sphere, arXiv:2605.12921 (2026)."
},
{
"url": "https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf"... | 2026-09-27T00:00:00 | Mark Hughes, Seungwon Kim, Maggie Miller, and Gheehyun Nahm, An irreducible real projective plane in the 4-sphere, arXiv:2605.12921 (2026).: https://arxiv.org/abs/2605.12921
Kirby et al., Problems in Low-Dimensional Topology, third-edition survey, Problem KP-4.37 (2026).: https://math.berkeley.edu/sites/default/files/s... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,914 | KP-4.38 | Kirby Problem 4.38 | Let $\Delta \subset B^{4}$ be a ribbon disk. Is $B^{4} \setminus \Delta$ aspherical, i.e. is $\pi_{i}(B^{4} \setminus \Delta)$ =0 for all i>1? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.38.
Literature notes:
(1) This is [Kir97, Problem 1.103]. Note that the Whitehead asphericity conjecture [Whi41, p. 428] (see also [Ros07a, How85] and Problem 5.11) predicts an affirmative answer. A generalized version, asking 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 |
2,915 | KP-4.39 | Kirby Problem 4.39 | Let K be a closed surface smoothly embedded in a 4-manifold $M^{4}$. Describe the subgroup of the mapping class group $Mod(K)$ of diffeomorphisms of K that extend to diffeomorphisms of M. Conversely, is it possible to characterize when a given subgroup of the mapping class group arises as the group of extendable diffeo... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.39.
Literature notes:
(1) The problem is due to Hirose, who has studied instances of this problem in e.g. [Hir93, Hir02, Hir05, Hir12]. Some highlights of his results on this problem are summarized in the remarks to follow.
(2) ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
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"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 |
2,916 | KP-4.40 | Kirby Problem 4.40 | Is every link of 2-spheres in $S^{4}$ slice? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.40.
Literature notes:
(1) A 2-link is slice if it bounds a disjointly embedded collection of 3-balls in the 5-ball $B^{5}$.
(2) Kervaire [Ker65a] proved that every 2-knot in $S^{4}$ is slice.
(3) This question can be asked in e... | 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",
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"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 |
2,917 | KP-4.41 | Kirby Problem 4.41 | Let $\Sigma$ be a compact surface and let X be a connected 4manifold. Let $f_{0}, f_{1}: \Sigma \to X$ be $\pi_{1}-negligible$ embeddings that agree on $\partial\Sigma$ and that are homotopic rel. boundary. Give computable invariants that decide whether there is $a \pi_{1}-negligible$ concordance in $X \times I$ betwee... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.41.
Literature notes:
(1) A surface is $\pi_{1}-negligible$ if the complement of the surface included into X induces an isomorphism on fundamental groups [FQ90]. This assumption is equivalent to asking for an immersed dual sphere... | 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 |
2,918 | KP-4.42 | Kirby Problem 4.42 | (a) If X is a smooth, closed, simply connected 4-manifold $withb_{2}(X) \geq$ 2, are all knots slice in X?
(b) In particular, are all knots slice in $\mathbb{CP}^{2}\#\overline{\mathbb{CP}}{}^{2}$? In the K3 surface?
(c) Can the set of slice knots detect exotic smooth structures? That is, do there exist homeomorphic ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.42.
Literature notes:
(1) A knot K is called slice in a closed 4-manifold X if it bounds a smoothly embedded disk in $X \setminus$ in $t(B^{4})$, where K lives on the boundary $S^{3}$. It is called H-slice if the disk is null-hom... | 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/10.1007/BF01389797",
"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 (b)**: K3 asks about CP² # CP²; the record prints CP² # −CP² (−CP² is CP² with reversed orientation, an overline in the source). K3 (2026), Problem 4.42 (p. 223, scribed by C. Manolescu) asks (a) whether all knots are slice in every... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,919 | KP-4.43 | Kirby Problem 4.43 | Let K be a knot on the boundary of $X \setminus B^{\circ 4}$, where X is a negative definite, smooth 4-manifold. Suppose K bounds a smoothly embedded, nullhomotopic surface $\Sigma$ in $X \setminus B^{\circ 4}$. Is the following inequality $2g(\Sigma) \geq$ |s(K)| (8) always satisfied? Here $s(K)$ denotes Rasmussen’s s... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.43.
Literature notes:
(1) For $X =S^{4}$, the inequality (8) was proved by Rasmussen [Ras10]. When $X =\#_{r}\mathbb{CP}^{2}$, (8) was proved by Manolescu–Marengon–Sarkar-Willis [MMSW23].
(2) Freedman–Gompf–Morrison–Walker [FGMW... | 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",
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"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 |
2,920 | KP-4.44 | Kirby Problem 4.44 | (a) Let X be a closed, simply connected 4-manifold. Let $g \geq$ 0 and $d \geq$ 0 be integers. Fix $x \in H_{2}(X;\mathbb{Z})$. Does there exist a (smooth, locally flat) generic immersion $f: \Sigma_{g} \to X$ such that $f_{*}[\Sigma_{g}] = x$, with d transverse double points?
(b) An important special case is $d =$ 0.... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.44.
Literature notes:
(1) The smooth genus function is known for $\mathbb{CP}^{2}$ by a celebrated result of Kronheimer–Mrowka [KM94]. The cases of $\mathbb{CP}^{2}\#\overline{\mathbb{CP}}{}^{2}$ and $S^{2} \times S^{2}$ were res... | 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 |
2,921 | KP-4.45 | Kirby Problem 4.45 | Let X be a closed simply connected smooth 4-manifold and let $\Sigma$ be a closed, orientable surface. Fix a smooth embedding of $f: \Sigma \hookrightarrow X$ to serve as a basepoint. For which $k\geq 0$ does the equality
$$
\ker i_{*}=\ker s_{*}
$$
hold, where
$$
\begin{aligned} i_{*}&:\pi_{k}(\operatorname{Emb}^{DIF... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.45.
Literature notes:
(1) Here $X^{\circ}$ denotes the punctured $X, X^{\circ} = X \setminus \operatorname{Int}(D^{4}), \operatorname{Emb}^{DIFF}(\Sigma,X^{\circ})$ is the space of smooth embeddings, $\operatorname{Emb}^{TOP}(\Si... | 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 |
2,922 | KP-4.46 | Kirby Problem 4.46 | Are all groups good? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.46.
Literature notes:
(1) The problem is due to Freedman and Quinn. The term ‘good’ group was first introduced in [Fre84] and [FQ90, p. 99] to refer to any group for which Freedman’s disk embedding theorem holds. In those two sou... | 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 |
2,923 | KP-4.47 | Kirby Problem 4.47 | (Round handle problem). Is there a link $L \subset S^{3}$ with vanishing pairwise linking numbers that is not round handle slice? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.47.
Literature notes:
(1) We define the terminology in the problem. A round handle is a copy of $D^{1} \times D^{2} \times S^{1}$, which is attached to the boundary of a 4-manifold along $S^{0} \times D^{2} \times S^{1}$. (So it ... | 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 |
2,924 | KP-4.48 | Kirby Problem 4.48 | Let $\Delta$ be a contractible, compact 4-manifold. Is the space of homeomorphisms of $\Delta$ that fix the boundary pointwise, with the compact-open topology, a contractible space? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.48.
Literature notes:
This is true for $\Delta = D^{4}$, by the Alexander trick. By PerronQuinn [Per86, Qui86], the space is connected. In dimensions at least 6, the space of homeomorphisms is contractible, as proven by Galatius–... | 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 |
2,925 | KP-4.49 | Kirby Problem 4.49 | Give an effective necessary and sufficient condition for an open 4-manifold to be homeomorphic to the interior of a compact 4-manifold. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.49.
Literature notes:
(1) In his thesis [Sie65], Siebenmann gave necessary and sufficient conditions for an n-dimensional open PL manifold M, for $n \geq$ 6, to be compactifiable by adding a manifold boundary (see also [BLL65]). ... | 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 |
2,926 | KP-4.50 | Kirby Problem 4.50 | Classify closed topological 4-manifolds (orientable and not) with finite fundamental group, up to homeomorphism. The following infinite families of fundamental groups are of particular interest.
(a) Dihedral groups.
(b) Quaternionic groups.
(c) Abelian groups with two or three generators. A significant starting poin... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.50.
Literature notes:
(1) The problem is deemed solved if the classification is reduced to algebraic invariants. An interesting subproblem is to complete the classification for smooth 4-manifolds only.
(2) The classification exi... | 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 |
2,927 | KP-4.51 | Kirby Problem 4.51 | Let X be a closed, smooth 4-manifold with fundamental group isomorphic to $\mathbb{Z}$. Is the $\mathbb{Z}[\mathbb{Z}]-valued$ intersection form on $\pi_{2}(X)$ extended from $\mathbb{Z}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.51.
Literature notes:
(1) We say that a Hermitian for m on a free $\mathbb{Z}[\mathbb{Z}]-module$ isextended from $\mathbb{Z}if$ it is represented by a matrix, with respect to some basis over $\mathbb{Z}[\mathbb{Z}]$, with intege... | 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 |
2,928 | KP-4.52 | Kirby Problem 4.52 | Let M and N be closed, orientable topological 4-manifolds with $\pi_{1}(M) \cong \pi_{1}(N)a$ good group. Suppose that M and N are simple homotopy equivalent and stably homeomorphic. Is there a simple homotopy equivalence $N \to M$ that lies in the orbit of Id $:M \to M$ under the Wall realization action of $L^{s}_{5}(... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.52.
Literature notes:
(1) We can hope to classify 4-manifolds up to simple homotopy equivalence, and up to stable homeomorphism, for many groups. We can also hope to compute $L^{s}_{5}(\mathbb{Z}[\pi_{1}(M)])$. If the answer were... | 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 |
2,929 | KP-4.53 | Kirby Problem 4.53 | Does there exist an algorithm that takes as input a closed, triangulated 4-manifold, and outputs in finite time whether or not that 4-manifold is homeomorphic to $S^{4}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.53.
Literature notes:
(1) This is called the recognition problem. We say that we can recognize an n-manifold X if there exists an algorithm that takes as input a closed, triangulated n-manifold, and outputs in finite time whether... | 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 |
2,930 | KP-4.54 | Kirby Problem 4.54 | The quadratic 2 type of a 4-manifold M is the data $(\pi_{1}(M), \pi_{2}(M), \lambda_{M}, k_{M})$ of the fundamental group $\pi_{1}(M)$, the second homotopy group $\pi_{2}(M)$ considered as $a \mathbb{Z}[\pi_{1}(M)]-module$, the equivariant intersection for $m \lambda_{M}$, and the k-invariant in $k_{M} \in H^{3}(B\pi_... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.54.
Literature notes:
(1) This is in [Kir97, Problem 4.1].
(2) We can also ask the question with the data of the Stiefel-Whitney classes $w_{1}$ and $w_{2}$. Which Stiefel-Whitney classes are realized within a given quadratic 2-... | 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 |
2,931 | KP-4.55 | Kirby Problem 4.55 | Let M and N be closed, orientable, connected 4-manifolds with isomorphic quadratic 2-types. If $\pi_{1}(M) \cong \pi_{1}(N)are$ finite, are M and N homotopy equivalent? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.55.
Literature notes:
(1) This is known for $\pi_{1}$ trivial, finite cyclic groups [HK88], dihedral groups [KPR24], and abelian groups with at most two generators [KNR22]. It is known to be false in the nonorientable case, due t... | 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 |
2,932 | KP-4.56 | Kirby Problem 4.56 | (4D s-cobordism conjecture). Let $(W^{4};M_{0}^{3}, M_{1}^{3})$ be a smooth 4-dimensionals-cobordism between closed 3-manifolds. Is W diffeomorphic to $M_{0} \times$ [0,1]? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.56.
Literature notes:
(1) Matumoto and Siebenmann found counterexamples to the topological analogue in [MS78], where both $M_{0}$ and $M_{1}$ are $\mathbb{RP}^{2} \times S^{1}$ and W is not known to be smoothable. Specifically th... | 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 |
2,933 | KP-4.57 | Kirby Problem 4.57 | Let X and Y be closed, oriented, smooth 4-manifolds with the same Euler characteristic and signature. Is there a torus link L in X with trivial normal bundle such that some choice of torus surgery along L transforms X into a manifold diffeomorphic to Y? What if we only require that the result be homeomorphic to Y? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.57.
Literature notes:
(1) These questions were discussed in [Ste06, BS13, FS11].
(2) Let T be a torus in a 4-manifold W with trivial normal bundle. We say that a 4-manifold $W^{1}is$ obtained from W by torus surgery along T if $... | 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] `10.1112/jtopol/jts038` is a paper by Ritter; cited paper: `10.1112/jtopol/jts027`. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,934 | KP-4.58 | Kirby Problem 4.58 | (a) Which Seifert fibered homology spheres $\Sigma(a_{1}$, . . . , $a_{n})bound$ acyclic manifolds? Are there any examples with four or more singular fibers that bound acyclic manifolds?
(b) Which Seifert fibered homology spheres bound contractible manifolds? Is there an example that bounds an acyclic manifold but not... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.58.
Literature notes:
(1) The second part of Problem (a) (regarding the number of fibers) is in [Kir97, Problem 4.123] in a different guise, with a different motivation.
(2) Many Seifert fibered homology spheres bound acyclic (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 |
2,935 | KP-4.59 | Kirby Problem 4.59 | Are lens spaces topologically homology cobordant if and only if they are homeomorphic? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.59.
Literature notes:
(1) Livingston conjectures that the statement is correct.
(2) If $L(a_{1}, b_{1})$ and $L(a_{2}, b_{2})$ are homology cobordant, then $a_{1} = a_{2}$. The problem can be restated: if L(n, $b_{1})$ and L(n, ... | 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 |
2,936 | KP-4.60 | Kirby Problem 4.60 | (a) Let X be an open, spin, smooth4-manifold. Does X have a proper smooth embedding in $\mathbb{R}^{6}$?
(b) By choosing a proper exhaustion function on W,(a) would follow from an affirmative answer to the following. Let (W;M, N) be a compact, smooth spin cobordism with a smooth embedding f of M in $S^{5}$. Is there a... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.60.
Literature notes:
(1) The spin condition is necessary in both parts of the problem.
(2) One motivation for part (a)is to understand theembedding dimension for a Stein surface (of real dimension four). By definition, this is ... | 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 |
2,937 | KP-4.61 | Kirby Problem 4.61 | What do different 4-manifold gauge theories see? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.61.
Literature notes:
(1) It is conjectured that the Donaldson, Seiberg–Witten, and Heegaard Floer invariants of closed 4-manifolds coincide (after organizing the Donaldson invariant as a generating function [KM95]). Yet each 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 | {
"source_url": "https://doi.org/10.1215/S0012-7094-95-07826-0",
"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: item (8) of the problem now has a Seiberg–Witten proof for cyclic groups of prime order (unrefereed preprint). D. Ruberman (K3 2026, Problem 4.61, pp. 239–240) asks what the different 4-manifold gauge theories see; item... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,938 | KP-4.62 | Kirby Problem 4.62 | Let X be a smooth, closed, connected, oriented 4-manifold with $b^{+}_{2}(X)$ >1.
(a) Does X have Donaldson simple type?
(b) Does X have Seiberg–Witten simple type? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.62.
Literature notes:
(1) The Simple Type Conjecture holds that the answer is affirmative. Part (a) is in [Kir97, Problem 4.131], and was raised by Kronheimer and Mrowka in [KM95] for simply connected 4–manifolds.
(2) A 4-manifo... | 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 |
2,939 | KP-4.63 | Kirby Problem 4.63 | How many independent basic classes can a simply connected smooth 4-manifold X have, as measured $bybr(X)$, the rank of the span of the basic classes?
(a) Is there an upper bound $forbr(X)$ in terms of topological invariants of X?
(b) In particular, is $br(X) \leq b^{+}_{2}(X)$ for all simply connected X with $b^{+}_{... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.63.
Literature notes:
(1) Here we define $br(X)=\operatorname{Rank}\operatorname{Span}\{c_{1}(\mathfrak{s})\mid \mathfrak{s}\text{ is a Seiberg-Witten basic class on }X\}$ if X has any basic classes, and 0 otherwise.
(2) Knot su... | 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 |
2,940 | KP-4.64 | Kirby Problem 4.64 | Find an irreducible, closed, smooth 4-manifold with nontrivial Bauer–Furuta invariant but with trivial Seiberg–Witten invariant. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.64.
Literature notes:
The Bauer–Furuta invariant $\Psi$ [BF04] is a stable cohomotopy refinement of the Seiberg–Witten invariant [Wit94], building on Furuta’s proof of the 10/8-theorem [Fur01]. There are examples for which the Ba... | 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 |
2,941 | KP-4.65 | Kirby Problem 4.65 | Suppose X is a smooth4-manifold with the homology of $S^{1} \times S^{3}$ whose infinite cyclic cover $\widetilde{X}$ has $H^{1}(\widetilde{X})=0$. Furuta and Ohta [FO93] define an invariant $\lambda_{FO}(X)$ as 1/4 of the signed count of irreducible flat $SU(2)$ connections on X; this requires an orientation of X and ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.65.
Literature notes:
(1) Part (a) was conjectured by Furuta–Ohta.
(2) A solution to (a) would imply that the Wall group $L_{5}(\mathbb{Z}[\mathbb{Z}])$ does not act on the smooth structure set of $S^{1} \times S^{3}$; compare t... | 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 |
2,942 | KP-4.66 | Kirby Problem 4.66 | Can the skein lasagna module detect exotic smooth structures on closed 4-manifolds? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.66.
Literature notes:
(1) The skein lasagna module is an extension of Khovanov homology. It is an invariant of 4-manifolds with (possibly empty) boundary and a framed link in their boundary. It was defined by Morrison, Walker 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 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,943 | KP-4.67 | Kirby Problem 4.67 | (a) Compute $\pi_{0}(\operatorname{Diff}^{+}(S^{4}))$. Do we have $\pi_{0}(\operatorname{Diff}^{+}(S^{4})) =$ \{1\}?
(b) In particular, does some implantation of the barbell map provide a nontrivial element in $\pi_{0}(\operatorname{Diff}^{+}(S^{4}))$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.67.
Literature notes:
(1) The 4-dimensional generalized Smale conjecture asked whether the inclusion $SO(5) \hookrightarrow \operatorname{Diff}^{+}(S^{4})$ is a homotopy equivalence (see Problems 4.34 and 4.126 in [Kir97]). Watan... | 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 |
2,944 | KP-4.68 | Kirby Problem 4.68 | Does every closed smooth 4-manifold admit an exotic diffeomorphism? How about the following special cases?
(a) Is there a definite smooth closed 4-manifold that admits an exotic diffeomorphism?
(b) Does $S^{2} \times S^{2}$ or K3 admit an exotic diffeomorphism? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.68.
Literature notes:
(1) A self-diffeomorphism f: $X \to X$ of a smooth manifold X is called exotic if it is topologically but not smoothly isotopic to the identity. The first examples of exotic diffeomorphisms of 4-manifolds we... | 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 |
2,945 | KP-4.69 | Kirby Problem 4.69 | Does there exist a diffeomorphism of a closed3-manifoldf: $M \to M$ such that f is topologically but not smoothly pseudo-isotopic to the identity? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.69.
Literature notes:
(1) Friedman–Witt [FW86] constructed diffeomorphisms f that are homotopic but not isotopic to the identity. These are also topologically pseudoisotopic to the identity [KS96].
\paragraph{Question.} Are the ... | 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 |
2,946 | KP-4.70 | Kirby Problem 4.70 | Do there exist $k \geq$ 0 and a smooth closed 4-manifold X such that the map $\pi_{k}(\operatorname{Diff}(X)) \to \pi_{k}(\operatorname{Homeo}(X))$ induced by the inclusion $\operatorname{Diff}(X) \hookrightarrow \operatorname{Homeo}(X)$ is an isomorphism? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.70.
Literature notes:
(1) Lin–Xie [LX23] proved that, for every orientable compact smooth 4manifold X, at least one of the following holds:
\noindent$\bullet$ $\pi_{1}(\operatorname{Diff}(X)) \to \pi_{1}(\operatorname{Homeo}(X))... | 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 |
2,947 | KP-4.71 | Kirby Problem 4.71 | (a) Do there exist $k \geq$ 0 and a smooth closed orientable 4-manifold X such that $\pi_{k}(\operatorname{Diff}(X))$ is finitely generated?
(b) Do there exist $k >$ 0 and a smooth closed orientable 4-manifold X such that $H_{k}(BDiff(X);\mathbb{Z})$ is finitely generated? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.71.
Literature notes:
(1) For each $k >$ 0, there exist simply-connected closed smooth 4-manifolds X where $H_{k}(BDiff(X);\mathbb{Z})$ are not finitely generated [Kon24b]. Also, Auckly–Ruberman [AR25] proved that, for eachk >0, ... | 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 |
2,948 | KP-4.72 | Kirby Problem 4.72 | Let X be a closed orientable topological 4-manifold with finite $\pi_{1}(X)$.
(a) Is $\pi_{k}(\operatorname{Homeo}(X))$ finitely generated for every $k \geq$ 0?
(b) Is $H_{k}(BHomeo(X);\mathbb{Z})$ finitely generated for every $k \geq$ 0? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.72.
Literature notes:
(1) For $\pi_{1}(X)$ =1, it follows from a result by Perron [Per86] and Quinn [Qui86] (cf. $[GGH^{+}23])$ that $\pi_{0}(\operatorname{Homeo}(X))$ is finitely generated, which implies that $H_{1}(BHomeo(X);\m... | 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 |
2,949 | KP-4.73 | Kirby Problem 4.73 | Does the Morlet correspondence $BDiff_{\partial}(D^{n}) \cong \Omega^{n}_{0}(Top(n)/O(n))$ (10) hold for n=4? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.73.
Literature notes:
(1) Here $BDiff_{\partial}(D^{n})denotes$ the classifying space of the diffeomorphism group of $D^{n}$ relative to its boundary. We write $Top(n)$ for the group of homeomorphisms on $\mathbb{R}^{n}$ that fix... | 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 |
2,950 | KP-4.74 | Kirby Problem 4.74 | Does there exist a closed, smooth 4-manifold X and a diffeomorphism f: $X \to X$ such that f is smoothly pseudo-isotopic to the identity, but f is not stably smoothly isotopic to the identity? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.74.
Literature notes:
(1) We can stabilize a diffeomorphism by making a choice of isotopy to one that is the identity on a 4-ball, connect summing with $\#_{k}S^{2} \times S^{2}$ using that 4-ball, for some k, and then extending ... | 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 |
2,951 | KP-4.75 | Kirby Problem 4.75 | Let X be a connected smooth 4-manifold with nonempty boundary, with finite $\pi_{1}(X)$, and let $k \geq$ 0. Let $Diff_{\partial}(X)$ denote the group of diffeomorphisms of X that are the identity near $\partial X$. Is the image of the natural map $s_{*}: \pi_{k}(Diff_{\partial}(X)) \to$ colim $\pi_{k}(Diff_{\partial}(... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.75.
Literature notes:
(1) Fixing a model of the interior connected sum $X\#S^{2} \times S^{2}$ defined by $X\#S^{2} \times S^{2}$ =X $\cup ((\partial X \times [0,1])\#S^{2} \times S^{2})$, we have a well-defined stabilization map... | 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 |
2,952 | KP-4.76 | Kirby Problem 4.76 | Let X be a closed, oriented, simply connected, smooth 4manifold and fix $k >$ 0. Is there $N \geq$ 0 such that, for every $n \geq N$, the natural map $\pi_{k}(\operatorname{Diff}^{+}(X\#_{n}S^{2} \times S^{2})) \to \pi_{k}(\operatorname{Homeo}^{+}(X\#_{n}S^{2} \times S^{2}))$ is surjective? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.76.
Literature notes:
(1) If $k =$ 0, the answer to the analogous question is affirmative. First, for N =2, it follows from Wall’s theorem [Wal64a] that the natural map $\pi_{0}(\operatorname{Diff}^{+}(X\#_{n}S^{2} \times S^{2}))... | 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 source proves only the analogue for k = 0 (Wall, Freedman, Quinn), and the question is posed for k > 0, where no case is decided. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,953 | KP-4.77 | Kirby Problem 4.77 | For which $k \geq$ 0 and closed smooth 4-manifold X does the equality $\ker(i_{*}: \pi_{k}(Diff_{\partial}(X^{\circ})) \to \pi_{k}(Homeo_{\partial}(X^{\circ}))) =\ker(s_{*}: \pi_{k}(Diff_{\partial}(X^{\circ})) \to$ colim $\pi_{k}(Diff_{\partial}(X^{\circ}\#_{n}S^{2} \times S^{2})))$ (13) $n \to \infty$ hold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.77.
Literature notes:
(1) Here $i: Diff_{\partial}(X^{\circ}) \to Homeo_{\partial}(X^{\circ})$ is the inclusion, and $s: Diff_{\partial}(X^{\circ}) \to Diff_{\partial}(X^{\circ}\#_{n}S^{2} \times S^{2})$ is the stabilization map ... | 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 |
2,954 | KP-4.78 | Kirby Problem 4.78 | Let X be a simply connected closed smooth 4-manifold. Does $BDiff(X)$ satisfy homological stability over $\mathbb{Q}$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.78.
Literature notes:
(1) As before, let $s: Diff_{\partial}(X^{\circ}) \to Diff_{\partial}(X^{\circ}\#S^{2} \times S^{2})$ be the stabilization map from Problem 4.75. The question asks whether, for eachk $\geq$ 0, the induced ma... | 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 |
2,955 | KP-4.79 | Kirby Problem 4.79 | Given $k >$ 0, is there a closed, simply connected, smooth 4manifold X and a nonzero homotopy class $\alpha \in \pi_{k}(Diff_{\partial}(X^{\circ}))$ such that $\alpha \in \ker(i_{*}: \pi_{k}(Diff_{\partial}(X^{\circ})) \to \pi_{k}(Homeo_{\partial}(X^{\circ})))$ (14) and $\alpha \notin \ker(s_{*}: \pi_{k}(Diff_{\partial... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.79.
Literature notes:
(1) Here we use the notation i, and s of Problem 4.75 and the previous two problems.
(2) The answer to an analogous statement for $k =$ 0 is known to be affirmative: Kronheimer–Mrowka [KM20] proved that the... | 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**: only the analogue for k = 0 is known (Kronheimer–Mrowka; J. Lin), and the question is posed for k > 0, where no such class is known. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,956 | KP-4.80 | Kirby Problem 4.80 | Is there $n >$ 2 and a smooth closed simply connected 4manifold X for which there is an element of ordernin the $subgroupker(\pi_{0}\operatorname{Diff}(X) \to \pi_{0}\operatorname{Homeo}(X))$? More generally, is there a subgroup of order nin $\ker(\pi_{0}\operatorname{Diff}(X) \to \pi_{0}\operatorname{Homeo}(X))$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.80.
Literature notes:
(1) An element of $\ker(\pi_{0}\operatorname{Diff}(X) \to \pi_{0}\operatorname{Homeo}(X))is$ called an exotic diffeomorphism. Examples of exotic diffeomorphisms by Ruberman [Rub98] and Baraglia–Konno [BK20] ... | 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 |
2,957 | KP-4.81 | Kirby Problem 4.81 | Is there a smooth, closed, simply connected 4-manifold X for which the group $\ker(\pi_{0}(\operatorname{Diff}(X)) \to \pi_{0}(\operatorname{Homeo}(X)))$ is finitely generated? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.81.
Literature notes:
(1) Let TDiff(X)be the group of diffeomorphisms that act trivially on $H_{*}(X;\mathbb{Z})$. Quinn [Qui86] and Perron [Per86] proved that, for $\pi_{1}(X) =$ \{1\}, the group $\ker(\pi_{0}(\operatorname{Diff... | 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 |
2,958 | KP-4.82 | Kirby Problem 4.82 | Let $\phi$ be a self-diffeomorphism of a closed, simply-connected, smooth 4-manifold X. Suppose that for every smooth surface $\Sigma$ in X, the surfaces $\Sigma$ and $\phi(\Sigma)$ are smoothly isotopic.
(a) Is $\phi$ necessarily smoothly isotopic to the identity map?
(b) Is $\phi smoothly$ isotopic to a diffeomorph... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.82.
Literature notes:
(1) In dimension three, a self-homeomorphism f of an irreducible 3-manifold $M^{3}$ that preserves free homotopy classes of loops is isotopic to the identity map [ABD+20]. The analogous result holds for simp... | 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 |
2,959 | KP-4.83 | Kirby Problem 4.83 | For which 4-manifolds does there exist a smooth structure such that there exists a non-smoothable homeomorphism with respect to that smooth structure? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.83.
Literature notes:
(1) We say that a homeomorphism is smoothableif it is rel. boundary isotopic to a diffeomorphism. If this does not hold, we say that the homeomorphism is non-smoothable.
(2) This question is interesting for... | 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 |
2,960 | KP-4.84 | Kirby Problem 4.84 | Is there a closed oriented smooth 4-manifold X for which every finite subgroup Gof the mapping class group $\pi_{0}(\operatorname{Diff}^{+}(X))can$ be realized by a finite group action on X? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.84.
Literature notes:
(1) This question relates to the study of the algebraic properties of diffeomorphism groups as opposed to their homotopy type. If there is a group-theoretic section $G \to \operatorname{Diff}^{+}(X)$ of the ... | 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 |
2,961 | KP-4.85 | Kirby Problem 4.85 | Is there a closed orientable smooth 4-manifold X for which the identity component $Diff_{0}(X)$ of the diffeomorphism group is not uniformly perfect? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.85.
Literature notes:
(1) By Mather and Thurston [Mat71, Mat74,Thu74a], the group $Diff_{0}(X)$ is perfect for every closed orientable manifold X.
(2) For a given group G, the commutator length of an element $g \in$ [G, G] is de... | 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 |
2,962 | KP-4.86 | Kirby Problem 4.86 | Is it the case that for every closed, smoothable topological 4manifold X, there exists a locally linear finite group action on X, such that for every smooth structure on X, the action is non-smoothable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.86.
Literature notes:
(1) Given a topological manifold X and a smooth structure $\mathcal{O}$ on X, we say that a locally linear topological $action\phi of a$ group on X isnon-smoothable with respect $to\mathcal{O} if\phi$ is not... | 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 |
2,963 | KP-4.87 | Kirby Problem 4.87 | Is there an exotic action of $\mathbb{Z}/n$ on $S^{4}$ with 0-dimensional fixed point set? 1-dimensional? 2-dimensional? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-4.87.
Literature notes:
(1) In the case that the fixed set is empty, any nontrivial symmetry is orientationreversing, son=2. In this case Cappell and Shaneson [CS76] constructed an exotic smooth $\mathbb{RP}^{4}$, whose double cove... | 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 exotic involutions of S⁴ are free (Cappell–Shaneson; Fintushel–Stern), and the knotted fixed 2-spheres of Giffen and Gordon are detected topologically; no exotic action with fixed set of dimension 0, 1 or 2 is known. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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