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Math-1811 · mathematics · Group theory

Questions in Geometric Group Theory — Q 10.2

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Statement

Definitions and mathematical statement

Let $X$ be a connected CW complex with no cells of dimension greater than three, and put $\pi=\pi_1(X)$. A local coefficient system of abelian groups on $X$ consists of groups transported isomorphically along paths, with transport depending only on endpoint-preserving homotopy and respecting composition. After choosing a basepoint, such a system is equivalent to a $\mathbb Z\pi$-module, with monodromy given by the fundamental-group action.

Its cohomology $H^3(X;\mathcal M)$ can be computed from the cellular chains of the universal cover using the corresponding module-valued cochains. Assume \[ H^3(X;\mathcal M)=0 \qquad\text{for every local coefficient system }\mathcal M. \] Does there exist an $X$ satisfying this condition which is not homotopy equivalent to any two-dimensional CW complex?

The target two-complex is allowed to have infinitely many cells, and the source wording does not require $X$ to be finite. The finite-complex version is an important additional restriction, but it is not silently substituted here. No asphericity assumption is made; higher homotopy in degree two can be present. Vanishing only for constant integer coefficients would be much weaker than the universal condition displayed above. Likewise, a particular three-cell presentation that cannot be collapsed directly does not constitute a counterexample, because an unrelated two-complex could still have the same homotopy type. The problem concerns homotopy dimension rather than a chosen cell decomposition.

Short English statement

Can a three-dimensional complex have vanishing third cohomology for every local system yet fail to have the homotopy type of a two-complex?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1811-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1811

The manuscript constructs a finite three-complex satisfying the local-coefficient cohomology condition and excludes every finite two-complex homotopy model. This page allows a two-complex with infinitely many cells, whose nonexistence is not claimed by the paper. Consequently the constructed example is not asserted to meet the target's stronger nonexistence requirement. It is a related result on the finite-target formulation.

Questions in Geometric Group Theory — Q 10.2

OpenAI's claim

Wall's finite D$2$ question is claimed to have a negative example: a finite three-dimensional complex with infinite fundamental group satisfies D$2$ but has no finite two-dimensional homotopy model. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a finite connected CW complex $X$ of dimension three such that, for its universal cover $\widetilde X$, \[ H_i(\widetilde X;\mathbb Z)=0\quad(i>2),\qquad H^3(X;M)=0\quad\text{for every }\mathbb Z[\pi_1X]\text{-module }M, \] yet $X$ is without a homotopy equivalent finite CW model in dimension two or below. The cohomology condition ranges over all local coefficient modules, and no asphericity assumption is made.

Outline of the argument

Manuscript 1 (family 321, 2026-10-06): A rank-one twisted-homology obstruction is proved by repeatedly lifting a fixed finite source through cyclic-cover image towers, whose simplex growth cannot continue. A five-generator presentation has a finitely normally generated perfect subgroup and a quotient involution detected by a complex character. Its explicit boundary matrix gives vanishing twisted second homology; finite plus construction preserves this while contradicting the necessary two-dimensional character condition.

Formal verification

Family 321 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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