# P3148: Whitehead asphericity conjecture

- ID: `P3148`
- Reference: `whitehead-asphericity`
- Page: https://theoremdb.org/statements/P3148
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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.

### Problem setup

- **Definition (aspherical).** A connected space with π_n=0 for every n≥2.
- **Definition (subcomplex).** A union of cells closed under taking their attaching boundaries.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Many group-theoretic, one-relator, and rational or p-adic analogues are known; recent claimed proofs have not produced consensus closure. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): 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](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): The general integral two-complex statement remains open in current peer-reviewed literature.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `whitehead-asphericity`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>A. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izvestiya: Mathematics 89 (2025). introduction. introduction https://www.mathnet.ru/links/4fbc817a2c66e119ba5013f03fd81599/im9597_eng.pdf
   - Also cited at A. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izvestiya: Mathematics 89 (2025). introduction
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Explicitly states the original conjecture remains open and develops analogues.
   - 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.
2. <a id="reference-2"></a>A review of Whitehead's asphericity, University of Waterloo notes (2024). Conjecture 1.1 and review https://www.math.uwaterloo.ca/~karigian/training/M04-pazmino-pullas-final-project.pdf
   - journal_article; secondary source; checked 2026-08-01
   - Source 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.