[#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?
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
Current 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]
1Records
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.
How the 4 records connect
ProblemWhitehead asphericity conjecture
2See also
How to cite
TheoremDB contributors, “Whitehead asphericity conjecture,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/whitehead-asphericityThis page as plain text: whitehead-asphericity.md
This problem includes 4 records joined by 3 typed links, sourced from mathnet.ru[1], current as of August 1, 2026.
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.