Variations on the Nerve Theorem

TL;DR

This paper generalizes the Nerve Theorem to broader classes of topological spaces and covers, utilizing the Čech complex, and applies these results to poset topology and fiber theorems.

Contribution

It extends the Nerve Theorem to CW complexes and open covers without local finiteness, and incorporates the multinerve for weaker hypotheses.

Findings

Extended Nerve Theorem to CW complexes and open covers.

Utilized Čech complex for analyzing covers.

Proved generalized crosscut and Poset Fiber Theorem variations.

Abstract

Given a locally finite cover of a simplicial complex by subcomplexes, Bj\"orner's version of the Nerve Theorem provides conditions under which the homotopy groups of the nerve agree with those of the original complex through a range of dimensions. We extend this result to covers of CW complexes by subcomplexes and to open covers of arbitrary topological spaces, without local finiteness restrictions. Moreover, we show that under somewhat weaker hypotheses, the same conclusion holds when one utilizes the multinerve introduced by Colin de Verdi\`ere, Ginot, and Goaoc. Our main tool is the \v{C}ech complex associated to a cover, as analyzed in work of Dugger and Isaksen. As applications, we prove a generalized crosscut theorem for posets and some variations on Quillen's Poset Fiber Theorem.