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arXiv · 2610.10074

Finite Chains of Two-Complexes and Acyclic Covers

Laurent Bartholdi·Roman Mikhailov
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Abstract

A negative answer to Whitehead's asphericity problem would follow from an infinite chain of two-complexes, starting with a non-aspherical one, in which every inclusion induces the zero map on second homotopy groups. We prove that a connected two-complex $K$ is the first term of such chains of every \emph{finite} length if and only if $K$ has a connected acyclic regular cover. In particular, there is a finite non-aspherical two-complex $K$, a presentation complex of $\mathrm{SL}(2,5)$, such that for every $n$ there is a strictly increasing chain of finite two-complexes $K=K_0\subset K_1\subset\dots\subset K_n$ in which all inclusions are zero on $π_2$.

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Laurent Bartholdi, Roman Mikhailov. 2026-10-07. Finite Chains of Two-Complexes and Acyclic Covers. https://arxiv.org/abs/2610.10074

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