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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,764 | KP-2.16 | Kirby Problem 2.16 | If two hyperbolic surfaces have the same unmarked simple
length spectra (i.e., the same multiset of lengths that correspond to simple closed
curves), are they isometric? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.16.
Literature notes:
(1) Let $M$ be a closed Riemannian manifold with negative sectional curva-
ture, and let $\mathcal{C}$ denote the set of free homotopy classes of closed curves in
$M$. In each homotopy class, there is a uniq... | 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": "2026-10-16",
"kind": "partial",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-16] **partially solved (yes for generic surfaces)**; the dataset currently labels it open. K3 Problem 2.16, pp. 97–98 (proposed by T. Aougab): if two hyperbolic surfaces have the same unmarked simple length spectra, are they isometric? H. Baik, I. Choi, D. M. Kim, Simple length... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,765 | KP-2.17 | Kirby Problem 2.17 | Suppose that $c$ is a geodesic current on a hyperbolic surface,
and suppose that, on the space of hyperbolic metrics on the surface, $c$ has the same
length function as a closed geodesic.
Must $c$ be a convex combination of closed
curves? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.17.
Literature notes:
(1) For an introduction to the theory of geodesic currents in general, see the
foundational work of Bonahon [Bon86, Bon88], as well as the contem-
porary book of Erlandsson–Souto [ES22].
(2) As an example, s... | 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 |
2,766 | KP-2.18 | Kirby Problem 2.18 | What is the best lower bound on the volume of a fibered hyper-
bolic 3-manifold one can give in terms of the translation length of the monodromy
with respect to various metric spaces, such as Teichmüller space with the Weil–
Peterson metric or the pants complex? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.18.
Literature notes:
(1) The monodromy of a hyperbolic 3–manifold that fibers over the circle is
a pseudo-Anosov homeomorphism, which acts with positive translation
length on several well-studied spaces, including Teichmüller sp... | 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 |
2,767 | KP-2.19 | Kirby Problem 2.19 | Which mapping classes give rise to arithmetic hyperbolic 3-
manifolds as their mapping tori? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.19.
Literature notes:
(1) Roughly speaking, arithmetic hyperbolic manifolds are a special class of
hyperbolic manifolds constructed by “number-theoretic” methods. They
enjoy many special properties, e.g., possessing a very rich g... | 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 |
2,768 | KP-2.20 | Kirby Problem 2.20 | Can one detect holomorphicity from a monodromy factoriza-
tion of a Lefschetz pencil, fibration, or a surface bundle over a surface? What are
the special properties of monodromy factorizations of holomorphic fibrations? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.20.
Literature notes:
(1) Unless $\ell= g -1 = 0$, any fibration corresponding to a monodromy factor-
ization as above can be made symplectic and $J$-holomorphic for an almost
complex structure $J$ compatible with the symplectic ... | 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/BFb0063355",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] keep **partially_solved**; update the literature note: besides necessary conditions there are sufficient criteria in low genus. İ. Baykur (K3 2026, Problem 2.20, p. 102) asks whether holomorphicity of a Lefschetz fibration, pencil or surface bundle can be detected from its ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,769 | KP-2.21 | Kirby Problem 2.21 | What is the minimum number, $m_{g,b}$, of right-handed Dehn
twists along essential curves into which the boundary multi-twist, $\Delta:= T_{\delta_{1}} \cdots T_{\delta_{b}}$,
can be factorized in $\operatorname{Mod}(S_{g}^{b})$? Find factorizations realizing the minimum. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.21.
Literature notes:
(1) A factorization $T_{c_1} \cdots T_{c_\ell}= \Delta$, where no $c_{i}$ is homotopic to a boundary
component or point, is often called a positive factorization of $\Delta$. A positive
factorization of the ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,770 | KP-2.22 | Kirby Problem 2.22 | Does every Lefschetz fibration over the 2–sphere admit a sec-
tion? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.22.
Literature notes:
(1) Here we assume the fibration has critical points; otherwise, there exist
$T^{2}$–bundles over $S^{2}$ without sections.
Lefschetz fibrations that arise from blowing up the base locus of Lef-
schetz penci... | 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/2610.01776",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **solved (no; unrefereed preprint)**. R. İ. Baykur (K3 2026, Problem 2.22, pp. 103–104) asks whether every Lefschetz fibration over S² (with critical points; not achiral) admits a section. R. İ. Baykur and N. Hamada, [arXiv:2610.01776](https://arxiv.org/abs/2610.01776) (1 O... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,771 | KP-2.23 | Kirby Problem 2.23 | Does there exist a surface bundle over a surface that admits
a complete hyperbolic metric, or, more generally, a complete metric of variable
negative curvature? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.23.
Literature notes:
(1) Thurston’s hyperbolization theorem provides for an enormous wealth of 3-
manifolds that are surface bundles over $S^{1}$, and that moreover admit com-
plete hyperbolic metrics. Moving up a dimension, 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**: Kent–Leininger's atoroidal bundles meet only a necessary condition; no bundle is known to be negatively curved, or even to have word-hyperbolic group, and no obstruction is known. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,772 | KP-2.24 | Kirby Problem 2.24 | Does there exist a complex surface $X$ that admits three or more
non-isomorphic structures as a surface bundle over a surface? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.24.
Literature notes:
(1) The analogous situation in the case of surface bundles over the circle is
organized by the Thurston norm [Thu86]: if a surface bundle over the
circle admits two different fibrations, then it admits infin... | 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 complex surfaces with exactly two fiberings are known, plus constraints on a third (Llosa Isenrich–Py); no example with three. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,773 | KP-2.25 | Kirby Problem 2.25 | Consider surface bundles over surfaces where both fiber $F$ and
base $B$ have genus $\geq 2$ and where $\pi_{1}(B)$ injects in the mapping class group of the
fiber. Does such a bundle have a $k$–multisection for some $k > 0$ (i.e., a continuous
choice of $k$ everywhere distinct points)? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.25.
Literature notes:
(1) This is Problem 2.17 in [Kir97] by G. Mess. In [Mor89], Morita gives
a cohomological obstruction for a surface bundle to admit a section (i.e.
the case $k = 1$). Using this obstruction, Hillman gave exam... | 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 results concern sections (k = 1): there are bundles without a section, which acquire one after a finite base change; the question about k-multisections is untouched. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,774 | KP-2.26 | Kirby Problem 2.26 | (Kontsevich–Zorich conjecture). Understand the homotopy types
of strata of abelian differentials. Which stratum-components are $K(\pi, 1)$ spaces?
What are the fundamental groups? For which stratum-components $\mathcal{H}$ is some ver-
sion of the monodromy map $\rho: \pi_{1}(\mathcal{H}) \to \operatorname{Mod}(S_{g})$... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.26.
Literature notes:
(1) A translation surface is a surface $T$ that admits an atlas of charts to $\mathbb{C}$
for which transition maps are translations ($z \mapsto z+c$). An abelian differ-
ential is a pair $(X, \omega)$ of 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 | {
"source_url": "https://doi.org/10.1007/s00222-003-0303-x",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **partially_solved** (π₁ and monodromy injectivity are decided for some strata; asphericity and the general case are open). N. Salter (K3 2026, Problem 2.26, author PDF pp. 106–107) asks, after Kontsevich–Zorich, which components of strata of abelian differentials are K(π, ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,775 | KP-2.27 | Kirby Problem 2.27 | For $n \geq 4$, does the braid group $B_{n}$ admit a finite-index sub-
group that embeds in a right-angled Artin group? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.27.
Literature notes:
(1) In CAT(0) geometry, a cube complex is called special if it satisfies certain
conditions on its hyperplanes [HW08]. Groups acting by combinatorial
isometries on special cube complexes are of particular in... | 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 |
2,776 | KP-2.28 | Kirby Problem 2.28 | Let $\Gamma$ be a graph that is not a nontrivial join, and let $A(\Gamma)$
be the associated right-angled Artin group. Does there exist an injective map from
$A(\Gamma)$ to the mapping class group of a surface such that every loxodromic element
of $A(\Gamma)$ is sent to a pseudo-Anosov mapping class? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.28.
Literature notes:
(1) There are many injective homomorphisms from right-angled Artin groups
into mapping class groups [CP01, CLM12, Kob12], and general criteria
are fairly well-understood [KK14b]. Right-angled Artin groups ha... | 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://zenodo.org/records/23248883",
"checked_at": "2026-10-08",
"kind": "claim",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 unrefereed claim, 2026-10-08] I have released an unrefereed preprint, **Sending Loxodromic Elements of Right-Angled Artin Groups to Pseudo-Anosov Mapping Classes: A Partial Answer to Problem 2.28 of the K3 List** (Alper Ferudun, Mercury Software GmbH). (https://zenodo.org/records/23248883) | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,777 | KP-2.29 | Kirby Problem 2.29 | Determine the Artin groups that can be embedded into a map-
ping class group. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.29.
Literature notes:
(1) If the Artin group is simply laced and the homomorphism sends each
standard generator to a Dehn twist, then there is an almost complete
classification. It is known that the Artin groups of type $A_{n}$ (... | 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 |
2,778 | KP-2.30 | Kirby Problem 2.30 | Which right-angled Artin groups contain closed hyperbolic sur-
face groups? Is there an algorithmic or graph-theoretic criterion to decide this? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.30.
Literature notes:
All but finitely many surface groups embed in right-angled Artin
groups [CW04], and right-angled Artin groups on cycles of length at least five
contain hyperbolic surface groups [SDS89]. Right-angled Artin g... | 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 |
2,779 | KP-2.31 | Kirby Problem 2.31 | Let $S$ be a closed surface of genus at least 2. Show that the
stable commutator length is rational on the commutator subgroup of $\pi_{1}(S)$. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.31.
Literature notes:
(1) If $G$ is a group and $g \in [G, G]$, the commutator length of $g$, denoted cl $(g)$,
is the least number of commutators in $G$ whose product is $g$, and the
stable commutator length of $g$, denoted scl ... | 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 |
2,780 | KP-2.32 | Kirby Problem 2.32 | Does every surface bundle over a surface admit a flat connec-
tion? What about surface bundles over 3-manifolds? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.32.
Literature notes:
(1) An $S_{g}$-bundle $p: E \to B$ admits a flat connection if it admits a codimen-
sion 2 foliation everywhere transverse to the fibers. Equivalently, the bun-
dle admits a flat connection if the monodromy ... | 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 |
2,781 | KP-2.33 | Kirby Problem 2.33 | Let $S$ be a closed compact surface (without boundary).
Is
there a finitely generated, torsion-free group $G$ such that $G$ cannot act faithfully by
homeomorphisms on $S$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.33.
Literature notes:
(1) There are non-compact surfaces for which every countable group acts by
homeomorphisms [APV21].
(2) We pose the question for closed surfaces only, since the presence of bound-
ary gives access to special ... | 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 |
2,782 | KP-2.34 | Kirby Problem 2.34 | Let $S$ be a compact surface. For $0 \leq r < s$, does there exist a
nontrivial finitely generated subgroup $G_{r} \leq \operatorname{Diff}^{r}_{0}(S)$ such that every action of $G_{r}$
on $S$ by $C^{s}$ diffeomorphisms is trivial? Can these examples be torsion-free? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.34.
Literature notes:
(1) For $0 \leq r \leq \infty$, let $\operatorname{Diff}^{r}_{0}(M)$ denote the identity component of the group
of $C^{r}$ diffeomorphisms of a manifold $M$.
For nonintegral $r$, we write
$r = k + \epsilon$ ... | 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 |
2,783 | KP-2.35 | Kirby Problem 2.35 | Is the first-order theory of the mapping class group of a surface
decidable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.35.
Literature notes:
(1) The first order theory of a group $G$ in the language of group theory is the
formal first-order language equipped with group multiplication and the
identity element. A first-order theory is decidable if ... | 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 |
2,784 | KP-2.36 | Kirby Problem 2.36 | Are systems of equations over mapping class groups and braid
groups decidable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.36.
Literature notes:
(1) A system of equations over a group $G$ is a finite conjunction of formulae
of the form
$$
r(x_{1},..., x_{n}, a_{1},..., a_{m}) = 1
$$
and
$$
r(x_{1},..., x_{n}, a_{1},..., a_{m}) \neq 1,
$$
where $r... | 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 |
2,785 | KP-2.37 | Kirby Problem 2.37 | Give a Nielsen–Thurston-type classification for the mapping
class groups of infinite-type surfaces. In particular, which homeomorphisms are the
appropriate generalization of pseudo-Anosov homeomorphisms in this setting? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.37.
Literature notes:
(1) For finite-type surfaces $S$, there is a powerful classification due to Nielsen
and Thurston of the mapping classes of $S$: every mapping class is periodic,
reducible, or pseudo-Anosov. The most importan... | 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 |
2,786 | KP-2.38 | Kirby Problem 2.38 | Give an appropriate analogue of the curve graph for infinite-
type surfaces, and characterize the surfaces for which no such graph exists. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.38.
Literature notes:
(1) A motivating philosophy in geometric group theory is that one can study
the algebra of a group $G$ via the geometry of the spaces on which $G$ acts
and the dynamics of such actions. When $G$ is the mappi... | 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 |
2,787 | KP-2.39 | Kirby Problem 2.39 | (a) Does the mapping class group of an infinite-genus surface with no planar
ends contain every countable group?
(b) Does the mapping class group of the one-ended infinite-genus surface have
any proper finite-index subgroups?
(c) For an infinite-type surface, describe and characterize the subgroup of
mapping classes wi... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.39.
Literature notes:
(1) The work of Aougab, Patel, and Vlamis [APV21] shows every mapping
class group of an infinite-genus surface with no planar ends contains every
finite group as a subgroup. Thus the problem is really about ... | 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 |
2,788 | KP-2.40 | Kirby Problem 2.40 | (a) Let $S$ be an infinite-type surface and $\varphi$ a mapping class for which there is
a (marked) conformal structure $\Sigma$ on $S$ with respect to which $\varphi$ is realized
as a quasi-conformal homeomorphism.
Then for each quasi-conformal
structure in the connected component of Teichmüller space containing $\Sig... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.40.
Literature notes:
(1) Let $S$ be a surface of infinite type, and let $\varphi$ be a mapping class for
which there is some (marked) conformal structure $\Sigma$ on $S$ for which $\varphi$
is realized by a quasiconformal homeo... | 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 |
2,789 | KP-2.41 | Kirby Problem 2.41 | Give a finite list of practically computable invariants of the
mapping class group or pure mapping class group of an infinite-type surface $S$ that
determine the topology of $S$. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.41.
Literature notes:
(1) Let $S$ be a surface, possibly of infinite type. When $S = S_{g,b}^{p}$ is of finite
type, the homeomorphism type of the surface is encoded in the isomor-
phism type of its mapping class group, with two ... | 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 |
2,790 | KP-2.42 | Kirby Problem 2.42 | Is the geodesic flow in almost every direction on the Chamanara
surface ergodic? What about on the translation surface considered by Bruin and
Lukina, which is similar to the Chamanara surface? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.42.
Literature notes:
(1) Translation surfaces have geodesic flows in every direction that preserve
Lebesgue measure. Kerckhoff, Masur, and Smillie proved that for every
finite-type translation surface, this is the only probabili... | 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 |
2,791 | KP-2.43 | Kirby Problem 2.43 | Let $X$ be a compact, totally disconnected subset of $\mathbb{R}^{2}$ with
$|X| \geq 2$, and let $\Gamma_{X}$ denote the mapping class group of $\mathbb{R}^{2} - X$.
(a) For $Y \subset X$, let $\Gamma_{X,Y}$ be the subgroup of $\Gamma_{X}$ permuting $Y$ . Classify in-
variants $q$ that are functorial.
(b) If $X \subset... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.43.
Literature notes:
(1) In Part (a), an invariant is functorial if for each $X$, there is a map
$q_{X}: \Gamma_{X} \to \mathbb{R}$ so that the following holds.
For each inclusion $Y \subset X$
and each $g \in \Gamma_{X,Y}$ with... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2609.17879",
"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 (c) is now answered for smooth maps in an unrefereed preprint. D. Calegari (K3 2026, Problem 2.43, author PDF pp. 121–123) asks five questions on the mapping class group Γ_X of the plane minus a compact totally dis... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,792 | KP-2.44 | Kirby Problem 2.44 | Given an infinite-type surface $S$, which homeomorphisms $f: S \to$
$S$ give rise to mapping tori $M_{f}$ that admit a hyperbolic structure? For those which
do admit a hyperbolic metric, is $M_{f}$ homeomorphic to the interior of a compact
hyperbolic manifold with totally geodesic boundary? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.44.
Literature notes:
(1) Thurston’s theorem on the hyperbolization of mapping tori states that
for a finite-type surface $S$ and a homeomorphism $f$ of the surface $S$, the
mapping torus
$$
M_{f} = S \times [0, 1]/(x, 0) \sim (... | 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 |
2,793 | KP-2.45 | Kirby Problem 2.45 | Compute the end-periodic cobordism group $\Delta^{e}_{2}$ of end-periodic
automorphisms (diffeomorphisms or homeomorphisms) of surfaces. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.45.
Literature notes:
(1) The (oriented) cobordism group of $n$-dimensional manifolds, denoted $\Delta_{n}$,
consists of equivalence classes of pairs $(M^{n}, f)$ where $M$ is a compact
oriented manifold and $f$ is a homeomorphis... | 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 |
2,794 | KP-2.46 | Kirby Problem 2.46 | (a) Which coarsely boundedly generated mapping class groups of infinite-type
surfaces are hyperbolic?
(b) Consider the class of surfaces with $n \geq 2$ ends, all accumulated by genus.
Are the mapping class groups of these surfaces quasi-isometric for different
values of $n$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.46.
Literature notes:
(1) Big mapping class groups are not finitely, countably, or compactly gener-
ated. There is a generalization of compactness called coarse boundedness,
defined and studied extensively by C. Rosendal [Ros13,... | 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 |
2,795 | KP-2.47 | Kirby Problem 2.47 | (a) Given a mapping class $\psi$ of a based surface $S$, there is an induced endo-
morphism of the symmetric product $\operatorname{Sym}^{i}(S)$ and hence an endofunctor
$\psi_{*}$ of the partially wrapped Fukaya category $\mathcal{F}(S)$ as constructed, say, by
Auroux [Aur10]. Compute the categorical entropy of $\psi_... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.47.
Literature notes:
(1) In (a), the categorical entropy is equal to
$$
\limsup_{n\to\infty}\left(\dim H_{*}(\widehat{\mathrm{CFDA}}(\psi^{n},i-g))\right)^{1/n},
$$
where $\widehat{\mathrm{CFDA}}(\psi)$ is the bordered Floer b... | 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 |
2,796 | KP-2.48 | Kirby Problem 2.48 | The mapping class group of a closed, orientable, genus $g$ sur-
face $S$ acts by symplectomorphisms on the symmetric product $\operatorname{Sym}^{g}(S)$. In partic-
ular, it acts smoothly on $\operatorname{Sym}^{g}(S)$.
(a) What is the kernel of the map from the mapping class group of $S$ to the
smooth mapping class gr... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.48.
Literature notes:
(1) The question relates to whether Heegaard Floer homology is truly a sym-
plectic invariant, or should be determined by the smooth isotopy classes
of the Heegaard tori. (Examples of the difference between ... | 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 |
2,797 | KP-2.49 | Kirby Problem 2.49 | (AMU conjecture). Let $S$ be a surface with negative Euler char-
acteristic. If $\varphi \in \operatorname{Mod}(S)$ acts by a pseudo-Anosov on some subsurface (including
$S$ itself), show that there exists $p_{0}(\varphi)$ such that the Witten–Reshetikhin–Turaev
representation $\rho_{p}(\varphi)$ has infinite order for... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.49.
Literature notes:
(1) Given a compact Lie group $G$, the Witten–Reshetikhin–Turaev $G$-quantum
representations, introduced by Witten [Wit89] and rigorously defined
by Reshetikhin and Turaev [RT91], are families of representat... | 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 |
2,798 | KP-2.50 | Kirby Problem 2.50 | (Volume conjecture for surface diffeomorphisms). Let $S$ be a
closed oriented surface, let $q=e^{2\pi i/n}$ be a root of unity, and let $\mathcal{K}^{q}(S)$ be the Kauff-
man bracket skein algebra. Let $\phi: S \to S$ be a pseudo-Anosov diffeomorphism with
mapping torus $M_{\phi}$, and let $r: \pi_{1}(S) \to \operatorn... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-2.50.
Literature notes:
(1) For a closed oriented surface $S$ and a root of unity $q$, results of [BW17a,
FKBL19, GJS25, KK22, FKBL25] more or less establish a one-to-
one correspondence between irreducible representations of the 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 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,799 | KP-3.1 | Kirby Problem 3.1 | Classify the smallest volume hyperbolic 3-manifolds of various types. In particular:
(a) Determine the nonorientable closed hyperbolic 3-manifolds of least volume.
(b) Determine the n-cusped hyperbolic 3-manifolds of least volume for each $n\geq 3$.
(c) Determine the smallest volume hyperbolic 3-manifolds with n ori... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.1.
Literature notes:
(1) Parts (a)--(c) appeared in [Kir97, Problem 3.60].
(2) There has been a lot of progress in determining the hyperbolic 3-manifolds of low volume [GHM+21]. In particular, the orientable hyperbolic manifold ... | 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 |
2,800 | KP-3.2 | Kirby Problem 3.2 | Show that the volumes of hyperbolic 3-manifolds are not all rationally related. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.2.
Literature notes:
(1) This was proposed by Thurston in [Thu82]. One could interpret "rationally related" to mean the volumes are all rational multiples of each other, or the weaker claim that they span a finite-dimensional $\m... | 3 | open | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": null,
"checked_at": "2026-10-03",
"kind": "open",
"evidence": "published"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **open**: no ratio of two hyperbolic volumes is known to be irrational; the irrationality of L(2, χ₋₃) (Calegari–Dimitrov–Tang) concerns a single number, not a ratio of volumes. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,801 | KP-3.3 | Kirby Problem 3.3 | Does every cusped hyperbolic 3-manifold have a geometric ideal triangulation? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.3.
Literature notes:
(1) A geometric ideal tetrahedron is the convex hull of any four points on the 3-sphere at infinity of $\mathbb{H}^3$ that do not all lie on a circle. Topologically, it is a tetrahedron with its vertices remo... | 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/jdg/1214441650",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **solved (yes; unrefereed preprint, proof checked by us)**. K3 2026, Problem 3.3 (p. 134, scribed by M. Lackenby) asks whether every cusped hyperbolic 3-manifold has a geometric ideal triangulation; the known obstacle was to subdivide the Epstein–Penner cells ([J. Different... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,802 | KP-3.4 | Kirby Problem 3.4 | (Chen--Yang Volume Conjecture). (a) Prove that, for any hyperbolic 3-manifold $M$,
$$
\lim_{\substack{r\to\infty\\ r\ \mathrm{odd}}}\frac{1}{r}\log\bigl(TV(M;e^{2\pi i/r})\bigr)
=\frac{1}{2\pi}\operatorname{Vol}(M).
$$
(b) Prove that, for any closed, oriented, hyperbolic 3-manifold $M$,
$$
\lim_{\substack{r\to\infty... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.4.
Literature notes:
(1) Given a root of unity $q$ so that $q^2$ is a primitive root of unity of odd order, $TV(M;q)$ denotes the corresponding Turaev--Viro invariant [TV92]. Similarly, $WRT(M;q)$ denotes the Witten--Reshetikhin-... | 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 |
2,803 | KP-3.5 | Kirby Problem 3.5 | (a) Do there exist closed non-Haken hyperbolic 3-manifolds with arbitrarily large injectivity radius?
(b) Does there exist a cofinal tower of regular covers of closed hyperbolic 3-manifolds where all of the manifolds in the tower are non-Haken?
(c) Is there a tower of hyperbolic rational homology spheres for which al... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.5.
Literature notes:
(1) A cofinal tower of regular covers is a sequence of covers
$$
\cdots \to M_2 \to M_1 \to M
$$
such that $\bigcap_i \pi_1(M_i)=1$ and $\pi_1(M_i)$ is a normal subgroup of $\pi_1(M)$ for all i.
(2) Questi... | 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**: Calegari–Dunfield's rational homology spheres of large injectivity radius are not known to be non-Haken, and no tower as in (b) or (c) is known. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,804 | KP-3.6 | Kirby Problem 3.6 | Given a cofinal tower of covers M $\leftarrow$ $M_1$ $\leftarrow$ $M_2$ $\leftarrow$ $\cdots$, is it true that the torsion subgroups $\operatorname{Tor}(M_{n})$ of $H_1(M_{n}, \mathbb{Z})$ and the hyperbolic volumes $\operatorname{vol}(M_{n})$ satisfy
$$
\lim_{n\to\infty}\frac{\log|\operatorname{Tor}(M_n)|}{\operatorn... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.6.
Literature notes:
(1) A tower of finite covers $M\leftarrow M_1\leftarrow M_2\leftarrow\cdots$ is called cofinal if the corresponding fundamental groups satisfy $\bigcap_n \pi_1(M_n)=\{1\}$.
(2) Bergeron and Venkatesh conject... | 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/10.1017/S1474748012000667",
"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: as literally posed (for every cofinal tower) the answer is no. K3 2026, Problem 3.6 (p. 136, scribed by D. Futer and M. Lackenby) asks whether log|Tor H₁(Mₙ; Z)| / vol(Mₙ) → 1/(6π) along a cofinal tower of covers; its r... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,805 | KP-3.7 | Kirby Problem 3.7 | Does every finite-volume hyperbolic 3-manifold admit a finitesheeted cover fibering over the circle with orientable pseudo-Anosov monodromy? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.7.
Literature notes:
(1) Agol, building on work of Wise and Kahn--Markovic, proved that every finite-volume hyperbolic 3-manifold admits a finite-sheeted covering that fibers over the circle [Ago08, Ago13, Wis21, KM12]. The monod... | 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 |
2,806 | KP-3.8 | Kirby Problem 3.8 | If $M_1$ and $M_2$ are finite-volume hyperbolic 3-manifolds whose fundamental groups have isomorphic profinite completions, must $M_1$ and $M_2$ be isometric? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.8.
Literature notes:
(1) The problem was stated by Reid in [Rei15, Question 9].
(2) Apart from its intrinsic interest, a positive answer to this problem would have some useful consequences. For example, it would provide a new so... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2608.07350",
"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: a full positive answer is now claimed in two unrefereed single-author preprints (not yet vetted). K3 2026, Problem 3.8 (p. 137, scribed by M. Lackenby; Reid's Question 9 of 2015) asks whether finite-volume hyperbolic 3-... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,807 | KP-3.9 | Kirby Problem 3.9 | Is being Haken a profinite invariant amongst 3-manifolds? That is, if $M_1$ and $M_2$ are 3-manifolds so that $\pi_1(M_{1})$ and $\pi_1(M_{2})$ have isomorphic profinite completions, and $M_1$ is Haken, must $M_2$ also be Haken? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.9.
Literature notes:
(1) The key case to consider is if M is a hyperbolic Haken rational homology sphere and N is a closed hyperbolic 3-manifold. A positive answer to Problem 3.8 implies a positive answer to this problem.
(2) No... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | {
"source_url": "https://arxiv.org/abs/2608.07350",
"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: for closed orientable 3-manifolds a positive answer follows from the claimed solution of `KP-3.8` (unrefereed preprints, not yet vetted). K3 2026, Problem 3.9 (p. 138, scribed by M. Lackenby, P. Patel and A. Reid) asks ... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,808 | KP-3.10 | Kirby Problem 3.10 | (a) Are there infinitely many commensurability classes of arithmetic rational homology 3-spheres?
(b) Are there infinitely many arithmetic integral homology 3-spheres? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.10.
Literature notes:
(1) Recall that two hyperbolic manifolds are commensurable if they have a common finite cover. Two lattices $\Gamma_1$ and $\Gamma_2$ in a Lie group G are commensurable if there is some $g\in G$ such that $\... | 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**: infinitely many arithmetic rational homology spheres are known, but not infinitely many commensurability classes, and only two arithmetic integral homology spheres are known. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,809 | KP-3.11 | Kirby Problem 3.11 | Does every hyperbolic knot in the 3-sphere have meridian length at most 4? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.11.
Literature notes:
(1) Every hyperbolic knot has a maximal cusp, the boundary of which is a Euclidean torus. The length of a slope on the knot exterior is the length of a geodesic representative of this slope in the Euclidean ... | 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 |
2,810 | KP-3.12 | Kirby Problem 3.12 | (a) Considering all closed, orientable, $\pi_1$-injective surfaces (possibly non-embedded) in all closed hyperbolic 3-manifolds, what is the infimum of the areas of all these surfaces?
(b) One can ask the same question, but considering just closed, orientable, $\pi_1$-injective surfaces of a fixed genus g.
(c) One ca... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.12.
Literature notes:
(1) A totally geodesic genus-two surface in a hyperbolic 3-manifold has area $4\pi$. A universal lower bound for orientable surfaces is $2\pi$ [Has95], based on unpublished work of Uhlenbeck. Hence, the answ... | 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 |
2,811 | KP-3.13 | Kirby Problem 3.13 | Does every closed hyperbolic 3-manifold admit an immersed $\pi_1$-injective surface with only double points? More precisely, if M is a closed, connected, hyperbolic 3-manifold, is there a closed, connected surface $\Sigma$ (other than a 2-sphere) and an immersion f : $\Sigma$ $\to$ M such that $f_*$ : $\pi_1(\Sigma)$ $... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.13.
Literature notes:
(1) This is true for Haken 3-manifolds, which by definition have embedded $\pi_1$-injective surfaces, and for small Seifert-fibered spaces with infinite fundamental group, which can easily be seen to have im... | 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 |
2,812 | KP-3.14 | Kirby Problem 3.14 | Can a hyperbolic knot complement in the 3-sphere contain a closed, embedded totally geodesic surface? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.14.
Literature notes:
(1) In 1990, Menasco and Reid conjectured that the answer is no [MR92]. There is evidence for and against this conjecture. For example, as evidence that it may be false, there are embedded totally geodesic s... | 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: Yes. DeBlois, Gharagozlou, and Hoffman construct four hyperbolic knot complements in S^3 containing closed embedded totally geodesic surfaces. This is a consistency correction based on that cited review; the discussion does not independently re-verify its pr... | [
{
"url": "https://arxiv.org/abs/2507.01263",
"label": "Jason DeBlois, Arshia Gharagozlou, and Neil R. Hoffman, Knot complements decomposing into prisms, arXiv:2507.01263 (2025)."
},
{
"url": "https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf",
"label": "R... | 2026-09-27T00:00:00 | Jason DeBlois, Arshia Gharagozlou, and Neil R. Hoffman, Knot complements decomposing into prisms, arXiv:2507.01263 (2025).: https://arxiv.org/abs/2507.01263
R. Inanc Baykur, Robion C. Kirby, and Daniel Ruberman, editors, K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (20... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,813 | KP-3.15 | Kirby Problem 3.15 | Let $M$ be a closed hyperbolic 3-manifold with positive first Betti number.
(a) Which elements of $H^{2}(M;\mathbb{R})$ are realized as the Euler classes of taut foliations?
(b) Which elements of $H^{2}(M;\mathbb{R})$ are realized as the Euler classes of tight contact structures?
(c) How about universally tight cont... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.15.
Literature notes:
(1) For any compact, orientable 3-manifold M, Thurston [Thu86] defined a pseudo-norm on the second homology groups $H_2(M;\mathbb{R})$ and $H_2(M,\partial M;\mathbb{R})$, now called the Thurston norm. The Th... | 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 |
2,814 | KP-3.16 | Kirby Problem 3.16 | Does every finite-volume hyperbolic 3-manifold contain infinitely many simple closed geodesics? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.16.
Literature notes:
It seems reasonable to suppose that, in some sense, a typical closed geodesic in a finite-volume hyperbolic 3-manifold does not self-intersect, and so one would expect the question to have a positive answer.... | 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/10.2140/agt.2006.6.2151",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **solved (yes; two independent unrefereed preprints)**. A. Reid (K3 2026, Problem 3.16, p. 142) asks whether every finite-volume hyperbolic 3-manifold contains infinitely many simple closed geodesics; the cusped case is Kuhlmann, [Algebr. Geom. Topol. 6 (2006)](https://doi.... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,815 | KP-3.17 | Kirby Problem 3.17 | Let $M_1$ and $M_2$ be finite-volume hyperbolic n--manifolds. If the length spectra of $M_1$ and $M_2$ coincide, must the two manifolds be commensurable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.17.
Literature notes:
(1) For a hyperbolic manifold M, the length spectrum $L(M)$ is the ordered tuple of all lengths of closed geodesics in M, listed with multiplicity. The length spectrum is closely related to the eigenvalue sp... | 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 |
2,816 | KP-3.18 | Kirby Problem 3.18 | Is there a closed hyperbolic 3-manifold that is foliated with minimal leaves? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.18.
Literature notes:
This problem was raised by Thurston and Uhlenbeck around 1980; see [Uhl83, Page 154]. Each leaf of such a foliation is an area minimizing surface, that is, any compact subsurface of a leaf minimizes area amo... | 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 special cases are excluded, namely minimal fibrations of certain fibered manifolds (Hass; Huang–Wang) and 'geometric' families of closed minimal surfaces (Wolf–Wu); there is no example and no general obstruction. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,817 | KP-3.19 | Kirby Problem 3.19 | (a) Does every closed hyperbolic 3-manifold have a nowhere zero vector field whose lift to the universal cover has proper flow lines?
(b) Can one ensure that the leaf space of the foliation by flow lines in the universal cover is Hausdorff ?
(c) If so, the leaf space in the universal cover is homeomorphic to a plane ... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.19.
Literature notes:
(1) The basic form of this question is due to W. Thurston.
(2) If the answer to part (c) is "yes," it seems likely one can show the action on the boundary must be faithful, in which case there are non-examp... | 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:math/9507216](https://arxiv.org/abs/math/9507216) is by S. Fenley and L. Mosher (cited: D. Calegari). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,818 | KP-3.20 | Kirby Problem 3.20 | Let $M$ be a closed hyperbolic 3-manifold with a faithful homomorphism $\rho:\pi_1(M)\to \operatorname{Homeo}^{+}(\mathbb{R})$. Prove that $M$ supports a dual transversely orientable, essential lamination $\lambda$. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.20.
Literature notes:
(1) Let $\rho$ be a faithful homomorphism as above. Suppose $\lambda$ is a lamination on $M$, and let $\widetilde{\lambda}$ be its lift to the universal cover. Then $\lambda$ is said to be dual to $\rho$ if ... | 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**: a transversely orientable dual lamination always exists, but no theorem makes it essential; Morgan–Shalen's R-tree analogue and universal circles from left orders are neighbouring results. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,819 | KP-3.21 | Kirby Problem 3.21 | In this problem, all 3-manifolds are orientable, while all flows are considered up to orbit equivalence and are assumed to be transitive.
(a) Are there only finitely many Anosov flows on a given closed 3-manifold?
(b) Are there only finitely many pseudo-Anosov flows on a given closed 3-manifold?
(c) Are there only f... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.21.
Literature notes:
(1) Two flows on a 3-manifold are orbit equivalent if there is an orientation-preserving homeomorphism of the manifold sending orbits of one flow to those of the other. A flow is transitive if it has a dense... | 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/2607.10398",
"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: parts (e) and (d) are now answered yes, and (a) for hyperbolic manifolds, all in unrefereed preprints. I. Agol (K3 2026, Problem 3.21, author PDF pp. 145–146) asks whether a closed 3-manifold has only finitely many tran... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,820 | KP-3.22 | Kirby Problem 3.22 | Let $G=\pi_1(M)$ be the fundamental group of a finite-volume hyperbolic 3-manifold $M$. What is the regularity of the smoothest (virtual) action of $G$ on $S^1$? On the interval? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.22.
Literature notes:
(1) Not all hyperbolic 3--manifold groups can act faithfully on the interval, as this is equivalent to left orderability. Calegari--Dunfield showed that the Weeks manifold group cannot act faithfully on the ... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,821 | KP-3.23 | Kirby Problem 3.23 | What is the Margulis constant in dimension 3? Is it realized uniquely by the Weeks manifold W, where $\mu(W)$ = 0.77442...? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.23.
Literature notes:
(1) The fundamental Margulis lemma in dimension 3 asserts that there is an $\epsilon>0$ so that for any discrete subgroup $\Gamma$ of $\operatorname{Isom}(\mathbb{H}^3)$, if $f,g\in\Gamma$ and
each move som... | 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 |
2,822 | KP-3.24 | Kirby Problem 3.24 | (a) (Cannon Conjecture) If G is a finitely presented, Gromov hyperbolic group with space at infinity equal to the 2-sphere, must G be a cocompact Kleinian group?
(b) More generally, if $G$ is a Gromov hyperbolic group whose boundary is planar, is G virtually a convex cocompact Kleinian group? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.24.
Literature notes:
(1) The first of these questions is Cannon's conjecture [Can94, CS98].
(2) Recall that a Gromov hyperbolic group G has a space at infinity $\partial_\infty G$, which is the space of geodesic rays based at s... | 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 |
2,823 | KP-3.25 | Kirby Problem 3.25 | (Bending Conjecture). (a) Is a quasi-Fuchsian group determined by the hyperbolic metric on the boundary of its convex core?
(b) Is a quasi-Fuchsian group with parabolics, other than a Fuchsian group, determined by the bending measure on the boundary of its convex core? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.25.
Literature notes:
Let $S$ be a closed surface (possibly with punctures) and let $\Gamma=\pi_1(S)$. Consider the space $QH(S)$ of quasi-Fuchsian representations, which corresponds to quasi-conformal deformations of Fuchsian re... | 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/10.1017/CBO9781139106993.005",
"checked_at": "2026-10-03",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-03] **partially_solved** ((b): yes, claimed in an unrefereed single-author preprint, not yet vetted; (a) is open). K3 (2026), Problem 3.25 (author PDF pp. 148–149, scribed by S. Maloni) asks, after Thurston, whether a quasi-Fuchsian group is determined (a) by the hyperbolic met... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,824 | KP-3.26 | Kirby Problem 3.26 | Let $M$ be a finite-volume hyperbolic 3-manifold, and let $M^1$ be a minimal-index finite cover of $M$ such that $\pi_1(M^1)$ embeds in a right-angled Artin group. Can one obtain good control over the complexity of the smallest right-angled Artin group containing $\pi_1(M^1)$ from the data of $M$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.26.
Literature notes:
(1) Work of Agol and Wise [Ago13, Wis21] established that there is always such a finite cover $M^1$.
(2) The complexity of a right-angled Artin group could be interpreted in many different ways, including:
... | 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 |
2,825 | KP-3.27 | Kirby Problem 3.27 | (a) What is the computational complexity of the homeomorphism problem for compact, orientable 3-manifolds?
(b) Is there a polynomial-time algorithm to recognize the 3-sphere?
(c) What is the computational complexity of the recognition problem for other 3-manifolds? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.27.
Literature notes:
(1) The homeomorphism problem for 3-manifolds is as follows. One is given triangulations of two 3-manifolds, M and N, and the problem asks whether N is homeomorphic to M.
(2) For a fixed 3-manifold M, its r... | 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 |
2,826 | KP-3.28 | Kirby Problem 3.28 | How many Pachner moves are needed to pass between two triangulations of a compact 3-manifold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.28.
Literature notes:
(1) Pachner [Pac91] proved that any two combinatorial triangulations of a compact piecewise-linear $n$-manifold differ by a finite sequence of modifications that are now called Pachner moves. In the case 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 |
2,827 | KP-3.29 | Kirby Problem 3.29 | (a) Given a closed hyperbolic 3-manifold $M$, can one find an explicit bound on the degree of a finite cover $\widetilde M$ having $b_1(\widetilde M)>0$?
(b) For example, is there such a finite cover with degree at most a polynomial function of the number of tetrahedra in some triangulation of $M$?
(c) Similarly, can... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.29.
Literature notes:
(1) Agol and Wise proved that $M$ has a finite cover $\widetilde M$ with $b_1(\widetilde M)>0$, and indeed a finite cover $\widetilde M_1$ that fibers over the circle [Ago13, Wis21]. However, the proof is no... | 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 |
2,828 | KP-3.30 | Kirby Problem 3.30 | (a) What is the computational complexity of determining whether a compact 3-manifold admits a hyperbolic structure?
(b) If a compact 3-manifold does admit a hyperbolic structure, what is the computational complexity of finding it?
(c) In particular, do these problems lie in the complexity classes NP and FNP respectiv... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.30.
Literature notes:
(1) The most efficient known algorithm to find a hyperbolic structure on a 3-manifold if one exists is due to Scull [Scu21], based on work of Kuperberg [Kup19a]. If the 3-manifold is given as a triangulation... | 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 |
2,829 | KP-3.31 | Kirby Problem 3.31 | Suppose M is a closed 3-manifold.
(a) Can one decide if the fundamental group of M is left-orderable?
(b) What is the complexity of a certificate of left-orderability, or of non-left-orderability? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.31.
Literature notes:
(1) A nontrivial group G is left-orderable if it admits a left-invariant total order.
(2) There is no algorithm to determine in general, from a finite presentation of a group G, whether G is left-orderable.... | 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 |
2,830 | KP-3.32 | Kirby Problem 3.32 | Is there an algorithm to determine whether two closed, embedded surfaces in $\mathbb{R}^3$ are isotopic? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.32.
Literature notes:
(1) The problem of deciding whether two incompressible surfaces in a compact orientable irreducible 3-manifold are isotopic is solvable, using work of Waldhausen [Wal68b, Proposition 5.4] (see also [Bar25, 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 |
2,831 | KP-3.33 | Kirby Problem 3.33 | Let M and N be closed orientable 3-manifolds. Prove that if there is a degree-1 map $f:M\to N$ then $g(M)\geq g(N)$, where $g(M)$ is the Heegaard genus of $M$. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.33.
Literature notes:
(1) This is a long-standing conjecture, dating back to Haken and Waldhausen [Hak66, Wal70]. The conjecture implies the Poincaré Conjecture: suppose a closed 3-manifold N is homotopy equivalent to $S^3$. Sinc... | 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 |
2,832 | KP-3.34 | Kirby Problem 3.34 | Do any two genus-g Heegaard splittings of a closed, orientable 3-manifold M become equivalent after at most g stabilizations? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.34.
Literature notes:
The answer "yes" was conjectured by W. Thurston. This has been studied extensively and upper bounds on the number of necessary stabilizations are known [Joh95], [RS96], but the conjectured optimal bound of g... | 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 |
2,833 | KP-3.35 | Kirby Problem 3.35 | Given a compact manifold M, let $r(M)$ denote the rank of its fundamental group and $g(M)$ denote its Heegaard genus.
(a) Does every closed orientable hyperbolic 3-manifold M with $r(M)$ = 2 satisfy $g(M)$ = 2?
(b) Is there a non-Haken manifold $M$ such that $r(M)<g(M)$?
(c) Is there a constant $k>0$ such that $g(M)... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.35.
Literature notes:
(1) For a closed orientable 3-manifold $M$, $r(M)$ is always at most $g(M)$. There are examples of Seifert fibered spaces where this inequality is strict [BZ84], and indeed the difference between Heegaard ge... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,834 | KP-3.36 | Kirby Problem 3.36 | (Simple loop conjecture) Let f : F $\to$ M be a 2-sided immersion of a surface into a 3-manifold such that $f_*$ : $\pi_1(F)$ $\to$ $\pi_1(M)$ is not injective. Must some simple closed curve lie in the kernel of $f_*$ ? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.36.
Literature notes:
This is [Kir97, Problem 3.96], proposed by J. Hass. When a simple closed curve lies in the kernel of $f_*$, $(F,f)$ is compressible to an immersion of a simpler surface. When f is an embedding, the conjectur... | 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 |
2,835 | KP-3.37 | Kirby Problem 3.37 | (a) Is every finitely generated 3-manifold group linear (over some field with characteristic zero)?
(b) If so, can one bound the dimension of a faithful linear representation? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.37.
Literature notes:
(1) Every finitely generated 3-manifold group is the fundamental group of a compact 3-manifold (possibly with boundary) [Sco73]. Any compact 3-manifold is finitely covered by a compact, orientable 3-manifold... | 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 |
2,836 | KP-3.38 | Kirby Problem 3.38 | Is every PD$_3$-group the fundamental group of a closed, aspherical 3-manifold? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.38.
Literature notes:
(1) This is [Kir97, Problem 3.77(A)], due to G. P. Scott, and is very closely related to the Cannon Conjecture (Problem 3.24). The more general question whether every PD$_n$-group is the fundamental group 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 |
2,837 | KP-3.39 | Kirby Problem 3.39 | (a) Is every finitely generated perfect group the normal closure of a single element?
(b) Is there an integral homology sphere whose fundamental group has weight bigger than one?
(c) For any $n$, is there an integral homology sphere whose fundamental group has weight bigger than $n$?
(d) Specifically, are the weight... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.39.
Literature notes:
(1) Part (a) is a longstanding question due to Wiegold; see [MK14, Problem 5.52]. Note added in proof: Chen and Lodha [CL25] have claimed a counterexample to the Wiegold problem.
(2) The weight of a group 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 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,838 | KP-3.40 | Kirby Problem 3.40 | Does every closed, orientable, hyperbolic 3-manifold admit a tight contact structure? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.40.
Literature notes:
(1) There exist closed, orientable contact 3-manifolds without tight contact structures, such as the connected sum $P\#\overline{P}$ of the Poincaré homology sphere with its orientation reversed [EH01b]. On ... | 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 |
2,839 | KP-3.41 | Kirby Problem 3.41 | Is it true that for every knot $K\subset S^3$, there is an integer $n_K$ such that $S^3_r(K)$ admits a tight contact structure for all $r\geq n_K$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.41.
Literature notes:
(1) This appears as Conjecture 4.5 (high surgery conjecture) in the notes for Stipsicz's 2010 ICM talk [Sti10], but Stipsicz reports that he learned of the question from Ko Honda.
(2) This question can be v... | 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 |
2,840 | KP-3.42 | Kirby Problem 3.42 | Does every tight contact 3-manifold have finite Giroux torsion? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.42.
Literature notes:
(1) An affirmative answer to the question would show that a contact manifold is tight if and only if it has finite torsion.
(2) Towards an affirmative answer, Gay proved in [Gay06] that strongly symplectica... | 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 |
2,841 | KP-3.43 | Kirby Problem 3.43 | Understand how various properties of contact structures behave under different kinds of symplectic cobordism. For instance:
(a) Is tightness preserved under exact symplectic cobordism?
(b) Is weak fillability preserved under weak symplectic cobordism?
(c) How do the support genus and Giroux torsion behave under Lege... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.43.
Literature notes:
(1) Wand showed that tightness is preserved by Legendrian surgery [Wan15]. It follows that tightness is preserved under Stein cobordism. Stein cobordisms are examples of exact symplectic cobordisms. It is kn... | 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 |
2,842 | KP-3.44 | Kirby Problem 3.44 | Is there an algorithm to decide, given an open book, whether the corresponding contact 3-manifold is tight or fillable? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.44.
Literature notes:
(1) See [Wan15] for work of Wand related to this problem.
(2) A similar question is whether there is an algorithm to decide, given some presentation of a 3-manifold, whether it admits a tight or fillable co... | 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 |
2,843 | KP-3.45 | Kirby Problem 3.45 | (a) Are there contact 3-manifolds with support genus greater than one?
(b) Are there contact 3-manifolds with arbitrarily large support genus? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.45.
Literature notes:
(1) The support genus of a contact 3-manifold $(Y,\xi)$ was defined by Etnyre-- Ozbagci [EO08] to be the minimum of
$$
\{\,g(S)\mid (S,\phi)\text{ is an open book supporting }(Y,\xi)\,\}.
$$
(2) Overtwiste... | 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 |
2,844 | KP-3.46 | Kirby Problem 3.46 | Let $\lambda$ be a contact form on a closed 3-manifold that is not a lens space. Must the associated Reeb flow have infinitely many simple periodic orbits? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.46.
Literature notes:
(1) On lens spaces, there are examples of contact forms with exactly two simple Reeb orbits; see [CGHHL23, Ex. 1.1]. Conversely, it is shown in [CGHHL23] that if a closed three-manifold admits a contact form... | 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 |
2,845 | KP-3.47 | Kirby Problem 3.47 | (a) Does every Reeb flow on $S^3$, associated to a contact form giving the standard contact structure, have an elliptic periodic orbit?
(b) What about if one restricts to Reeb flows on $S^3$ arising from boundaries of strictly convex domains in $\mathbb{R}^4$ ? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.47.
Literature notes:
(1) It is known that no flow on $S^3$ can be Anosov.
(2) For some dynamical implications of the existence of an elliptic orbit; see [New77].
(3) For more about what is known the problem, including in highe... | 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 |
2,846 | KP-3.48 | Kirby Problem 3.48 | (L-space Conjecture). For prime rational homology 3-spheres Y, are the following equivalent?
(a) $\pi_1(Y)$ is left-orderable.
(b) Y is not an L-space.
(c) Y admits a coorientable taut foliation. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.48.
Literature notes:
(1) This problem is known as the L-space Conjecture. The question of whether
(b) is equivalent to (c) was first raised by Ozsváth--Szabó following their proof (completed by Bowden and Kazez--Roberts) that (... | 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 |
2,847 | KP-3.49 | Kirby Problem 3.49 | Are any of the three conditions in the L-space Conjecture equivalent, for all prime rational homology 3-spheres Y, to the condition that Y admits a contact structure that is filled by a symplectic 4-manifold with $b_2^+>0$. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.49.
Literature notes:
In proving that the existence of a coorientable taut foliation on Y implies that Y is not an L-space, Ozsváth--Szabó construct [OS04b] a weak symplectic filling of a particular contact structure on Y. This 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 |
2,848 | KP-3.50 | Kirby Problem 3.50 | (The Floer Poincaré Conjecture). If Y is an integral homology sphere that is an L-space, show that Y is $S^3$ or the connected sum of some copies of the Poincaré sphere (with arbitrary orientations). | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.50.
Literature notes:
(1) This is in the spirit of [Kir97, Problem 3.106], which asks a similar question about instanton Floer homology.
(2) The statement is already known to be true for several classes of homology spheres: Dehn... | 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 |
2,849 | KP-3.51 | Kirby Problem 3.51 | Suppose Y is a rational homology 3-sphere such that every homomorphism $\pi_1(Y)$ $\to$ $\operatorname{SU}(2)$ has abelian image. Does it follow that
$$
\dim_{\mathbb{C}} I^{\#}(Y;\mathbb{C})=|H_1(Y;\mathbb{Z})|,
$$
i.e., that Y is an instanton L-space? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.51.
Literature notes:
(1) The conclusion follows if the corresponding set of reducible flat connections is Morse--Bott nondegenerate for the Chern--Simons functional, which is equivalent to Y being cyclically finite [BN90, BS18b]... | 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 |
2,850 | KP-3.52 | Kirby Problem 3.52 | (a) Does every closed 3-manifold M besides the 3-sphere admit a nontrivial representation $\pi_1(M)$ $\to$ $\operatorname{SU}(2)$?
(b) For which M with nonabelian fundamental group does every homomorphism $\pi_1(M)$ $\to$ $\operatorname{SU}(2)$ have abelian image? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.52.
Literature notes:
(1) The question in (a) is [Kir97, Problem 3.105(A)].
(2) The answer to (a) is yes if $M\not\cong S^3$ is Seifert fibered [FS90] or toroidal [LPCZ23] (see also [BS22]), leaving only the hyperbolic case open... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,851 | KP-3.53 | Kirby Problem 3.53 | Are all strong L-spaces branched double covers of alternating links in $S^3$? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.53.
Literature notes:
(1) A rational homology 3-sphere $Y$ is an L-space if
$$
\operatorname{rank}\widehat{HF}(Y)=|H_1(Y)|,
$$
where $|H_1(Y)|$ denotes the number of elements in $H_1(Y)$. A strong L-space is a rational homology... | 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 |
2,852 | KP-3.54 | Kirby Problem 3.54 | (a) Is there a closed 3-manifold $M$ whose Heegaard Floer homology $\widehat{HF}(M;\mathbb{Z})$ has torsion?
(b) Is there a rational homology 3-sphere $M$ for which $\widehat{HF}(M;\mathbb{Z})$ has torsion?
(c) Is there a knot $K\subset S^3$ whose knot Floer homology $\widehat{HFK}(S^3,K;\mathbb{Z})$ has torsion? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.54.
Literature notes:
(1) The Heegaard Floer homology of a closed 3-manifold $M$ with integer coefficients depends on a choice of isomorphism class of coherent orientation systems (and similarly for the knot Floer homology of lin... | 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 torsion is in HF⁺ and HF^∞ (Jabuka–Mark), for a non-canonical orientation system, or in instanton homology; none is in the groups asked about in (a)–(c). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,853 | KP-3.55 | Kirby Problem 3.55 | (a) For $K$ a nontrivial knot in $S^3$, does $HFK^-(K)$ always admit an $\mathbb{F}_2$-summand, as an $\mathbb{F}_2[U]$-module?
(b) For $Y$ a rational homology sphere, if $HF_{\mathrm{red}}(Y)$ is nontrivial, does $HF_{\mathrm{red}}(Y)$ always admit an $\mathbb{F}_2$-summand? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.55.
Literature notes:
(1) The answer to Part (a) is "yes" for fibered knots [BVV18] and more generally for knots with $\widehat{HFK}(K,g)$ supported in a single $\mathbb{Z}/2\mathbb{Z}$-grading [Ni22]. See also [HW18].
(2) Regar... | 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 |
2,854 | KP-3.56 | Kirby Problem 3.56 | (a) For $Y^3$ a rational homology sphere, is the Seiberg--Witten Floer spectrum $SWF(Y)$ always a wedge of spheres?
(b) Is every monopole Floer homology L-space a Seiberg--Witten Floer homotopy L-space?
(c) Find an example of two rational homology spheres with the same monopole Floer homology but with different Seibe... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.56.
Literature notes:
(1) There are some explicit computations of $SWF(Y)$. In certain cases, the critical points and gradient trajectories of the Chern--Simons--Dirac functional can be identified explicitly by work of Mrowka, Oz... | 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 |
2,855 | KP-3.57 | Kirby Problem 3.57 | Construct an $S^1$ - or $\operatorname{Pin}(2)$-equivariant lattice homotopy type that computes the Seiberg--Witten Floer homotopy type. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.57.
Literature notes:
Lattice homology is an invariant for plumbed 3-manifolds introduced by Némethi [Ném05, Ném08], based on earlier work by Ozsváth and Szabó [OS03b]. Zemke has announced that lattice homology is isomorphic to H... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,856 | KP-3.58 | Kirby Problem 3.58 | Prove that there is an isomorphism relating Heegaard Floer homology and monopole Floer homology that commutes with the cobordism maps in the two settings. In particular, prove that the mixed invariant in Heegaard Floer homology is equal to the Seiberg--Witten invariant for closed 4-manifolds. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.58.
Literature notes:
As discussed in the section introduction, the Heegaard Floer homology of a closed 3-manifold is isomorphic to its monopole Floer homology, but it is not known whether these isomorphisms respect the TQFT-like... | 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 |
2,857 | KP-3.59 | Kirby Problem 3.59 | Prove an isomorphism relating instanton Floer homology and Heegaard Floer homology. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.59.
Literature notes:
(1) There is some context for this question in the section introduction.
(2) Kronheimer--Mrowka conjectured that sutured instanton homology is isomorphic to sutured Heegaard Floer homology with coefficients... | 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 |
2,858 | KP-3.60 | Kirby Problem 3.60 | Find an algorithm to compute instanton Floer homology of closed 3-manifolds and the Donaldson invariants of closed 4-manifolds. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.60.
Literature notes:
There are algorithms to compute the Heegaard Floer invariants of knots [MOS09], 3-manifolds [SW10, MOT25], cobordisms [LMW08, MOT25], and the Heegaard Floer mixed invariant of closed 4-manifolds [MOT25]. Per... | 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 |
2,859 | KP-3.61 | Kirby Problem 3.61 | Is the dimension of Heegaard Floer homology invariant under genus 2 mutation? | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.61.
Literature notes:
(1) Let $\Sigma$ be a closed, genus 2 surface embedded in a 3-manifold $M$. The genus 2 mutant $M_\Sigma$ is the 3-manifold obtained by cutting $M$ along $\Sigma$ and regluing by the hyperelliptic involution... | 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 |
2,860 | KP-3.62 | Kirby Problem 3.62 | How do Floer homological invariants behave under maps of nonzero degree? For instance, let $Y$ and $Z$ be closed, oriented 3-manifolds, and suppose that there is a degree-1 map $f:Y\to Z$. Is it true that
\begin{equation}\label{eq:hf-rank}
\operatorname{rk} HF(Y)\geq \operatorname{rk} HF(Z),
\tag{6}
\end{equation}
wher... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.62.
Literature notes:
(1) This question is inspired by the fact that many 3-manifold invariants, from the first Betti number to simplicial volume, exhibit monotonic behaviors under maps of nonzero degree; see [Wan02]. Perhaps mor... | 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 |
2,861 | KP-3.63 | Kirby Problem 3.63 | Give a method for computing the $\eta$ invariant for the Dirac operator, $\eta_{\mathrm{Dirac}}(Y, s)$, associated to a spin structure s on a hyperbolic 3-manifold Y. | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.63.
Literature notes:
(1) The Atiyah--Patodi--Singer $\eta$-invariant [APS75a, Goe12] is a real number defined in terms of the spectrum of a first-order self-adjoint elliptic differential operator A on a closed odd-dimensional or... | 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 |
2,862 | KP-3.64 | Kirby Problem 3.64 | Let A be a flat connection on the trivial $\operatorname{SU}(2)$ bundle on a closed three-manifold M. The Chern--Simons invariant $\operatorname{CS}(M, A)$ is defined to be
$$
\operatorname{CS}(M,A)=\frac{1}{4\pi^2}\int_M \operatorname{Tr}\!\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right).
$$
Is $\operatorname{CS... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.64.
Literature notes:
(1) This was asked in [Auc94], and implicitly in [KK90]. It is related to Problems 3.62, 3.63, and 3.104 in [Kir97].
(2) Rationality holds for all Seifert fibered spaces and graph manifolds [FS90, Fur90, Au... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,863 | KP-3.65 | Kirby Problem 3.65 | Let $S_{2,\infty}(Y)$ denote the Kauffman bracket skein module of a closed, oriented 3-manifold $Y$; this is a module over $R=\mathbb{Z}[A,A^{-1}]$. For $Y$ prime and connected, does $S_{2,\infty}(Y)$ having rank one, in the sense that
$$
S_{2,\infty}(Y)\otimes_R \mathbb{Q}(A)\cong \mathbb{Q}(A),
$$
imply that $Y$ is... | Source: Kirby's Problems in Low-Dimensional Topology notes. Original problem number: KP-3.65.
Literature notes:
(a) This problem appears as [DKS25, Question 10.3].
(b) Przytycki [Prz00] showed that $S_{2,\infty}(Y)\otimes_R \mathbb{Q}(A)$ is multiplicative under connected sum. Hence, the set of (closed, connected) 3-... | 3 | partially_solved | null | null | 7 | 11 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 11,
"name": "kirby_low_dimensional_topology",
"display_name": "Kirby's Problems in Low-Dimensional Topology",
"description": "Problems from Kirby-style notes on low-dimensional topology, including knot theory, 3-manifolds, 4-manifolds, and related invariants.",
"slug": "kirby-low-dimensional-topology"... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
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