[#P3148] Whitehead asphericity conjecture

Problem. If \(X\) is an aspherical connected two-dimensional CW complex and \(Y\subset X\) is a connected subcomplex, must \(Y\) also be aspherical?

A subcomplex inside an aspherical two-complex. — A structural topology diagram of the statement's mathematical objects.

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Definitions and notation

1Context

Known frontier: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Open boundary: The general integral two-complex statement remains open in current peer-reviewed literature.

2Problem setup

Definition 1 (aspherical). A connected space with π_n=0 for every n≥2.

Definition 2 (subcomplex). A union of cells closed under taking their attaching boundaries.

Remark 1. Asphericity means the universal cover is contractible, equivalently π_n vanishes for n≥2 in this two-dimensional setting. The conjecture links low-dimensional topology, group cohomology, and the Eilenberg-Ganea problem.

3What counts as a solution

Prove every connected subcomplex Y is aspherical.
Or give an explicit aspherical two-complex with a connected subcomplex having nonzero π₂.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Exact unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Exact unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Exact unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Whitehead asphericity conjecture. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. The exact unresolved remainder is: The general integral two-complex statement remains open in current peer-reviewed literature. A complete resolution must meet the following acceptance conditions: - Prove every connected subcomplex Y is aspherical. - Or give an explicit aspherical two-complex with a connected subcomplex having nonzero π₂. Open full record →
Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure.^[1][2] Relevance to this problem Locates the present research frontier immediately below Whitehead asphericity conjecture. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. This leaves the following boundary unresolved: The general integral two-complex statement remains open in current peer-reviewed literature. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.^[1][2] Relevance to this problem Documents why Whitehead asphericity conjecture was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Whitehead asphericity conjecture still open 2025 - aspherical 2 complex subcomplex conjecture claimed proofs status The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature. Related: Claim 1 Open full record →
The general integral two-complex statement remains open in current peer-reviewed literature.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 The general integral two-complex statement remains open in current peer-reviewed literature. Relevance to this problem Turns the remaining uncertainty in Whitehead asphericity conjecture into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: The general integral two-complex statement remains open in current peer-reviewed literature. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove every connected subcomplex Y is aspherical. - Or give an explicit aspherical two-complex with a connected subcomplex having nonzero π₂. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. Exact unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.

Equivalent-formulation queries: Whitehead asphericity conjecture still open 2025; aspherical 2 complex subcomplex conjecture claimed proofs status
Strongest checked neighboring result: Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure.
Exact unresolved remainder: The general integral two-complex statement remains open in current peer-reviewed literature.

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ProblemWhitehead asphericity conjecture

All 3 recorded relations between these records and the problem

2See also

Torsion-freeness for king-grid independence complexestopology
Integral torsion in scale-four hypercube Rips complexestopology
Hodge conjecturetopology

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Plain text

“Whitehead asphericity conjecture.” TheoremDB. P3148. Problem statement; statement text SHA-256 da4081c41257058e877f8062c3b76ff0a157e9458ad2fcd5b2d07ce04051e8fa. https://theoremdb.org/statement/?ref=P3148

BibTeX

@misc{theoremdb-problem-da4081c41257058e877f8062c3b76ff0a157e9458ad2fcd5b2d07ce04051e8fa,
  title = {{Whitehead asphericity conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 da4081c41257058e877f8062c3b76ff0a157e9458ad2fcd5b2d07ce04051e8fa},
  url = {https://theoremdb.org/statement/?ref=P3148}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from mathnet.ru^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. A. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izvestiya: Mathematics 89 (2025). introduction. introduction. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Explicitly states the original conjecture remains open and develops analogues.Also cited at A. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izvestiya: Mathematics 89 (2025). introduction.Source used to assess the problem's recorded status.For Whitehead asphericity conjecture: This is the dated publication status for the canonical target Whitehead asphericity conjecture.Source named by the research packet.
A review of Whitehead's asphericity, University of Waterloo notes (2024). Conjecture 1.1 and review. ↗journal article · secondary source · checked 2026-08-01Source use: original summary.Provides a modern self-contained review of formulations and partial results.Source used to assess the problem's recorded status.For Whitehead asphericity conjecture: Provides a modern self-contained review of formulations and partial results.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3148] Whitehead asphericity conjecture

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3What counts as a solution

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