PreprintCC BY 4.0arXiv
Finite Chains of Two-Complexes and Acyclic Covers
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 SL(2,5), such that for every n there is a strictly increasing chain of finite two-complexes K=K0⊂ K1⊂…⊂ Kn in which all inclusions are zero on π2.
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BibTeX
@misc{bartholdi2026finite,
title = {{Finite Chains of Two-Complexes and Acyclic Covers}},
author = {Laurent Bartholdi and Roman Mikhailov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10074},
eprint = {2610.10074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10074},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite Chains of Two-Complexes and Acyclic Covers
AU - Laurent Bartholdi
AU - Roman Mikhailov
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10074
UR - https://arxiv.org/abs/2610.10074
AB - 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 SL(2,5), such that for every n there is a strictly increasing chain of finite two-complexes K=K_0⊂ K_1⊂…⊂ K_n in which all inclusions are zero on π_2.
ER -