有限二复形链与无环覆盖
Finite Chains of Two-Complexes and Acyclic Covers
浏览论文内容
中文总结 AI 辅助
本文证明连通二复形是所有有限长度零映射链的首项当且仅当它有无环正则覆盖，并给出SL(2,5)表示复形的实例。
中文摘要 AI 辅助
怀特海德非球面性问题的否定答案将源于一个无限的二复形链，该链从一个非球面的二复形开始，其中每个包含映射在第二同伦群上诱导零映射。我们证明，一个连通的二复形 $K$ 是这种所有有限长度链的第一项，当且仅当 $K$ 具有连通的、无环的正则覆盖。特别地，存在一个有限非球面的二复形 $K$，它是 $\mathrm{SL}(2,5)$ 的一个表示复形，使得对于每个 $n$，存在一个严格递增的有限二复形链 $K=K_0\subset K_1\subset\dots\subset K_n$，其中所有包含映射在 $\pi_2$ 上为零。
英文摘要
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 $π_2$.
