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Math-3030 · mathematics · Topology

Wall D(2) problem with a finite target model

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $X$ be a finite connected three-dimensional CW complex and $\widetilde X$ its universal cover. If $H_i(\widetilde X;\mathbb Z)=0$ for $i>2$, and $H^3(X;M)=0$ for every $\mathbb Z[\pi_1X]$-module $M$, must $X$ be homotopy equivalent to a finite CW complex of dimension at most two?

Short English statement

Let $X$ be a finite connected three-dimensional CW complex and $\widetilde X$ its universal cover. If $H_i(\widetilde X;\mathbb Z)=0$ for $i>2$, and $H^3(X;M)=0$ for every $\mathbb Z[\pi_1X]$-module $M$, must $X$ be homotopy equivalent to a finite CW complex of dimension at most two?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3030-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3030

Manuscript 1 (family 321, 2026-10-06): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Wall D$2$ problem with a finite target model

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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