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Variations on the Nerve Theorem

Daniel A. Ramras

TL;DR

This work broadens the classical Nerve Theorem to covers of CW complexes and open covers of arbitrary spaces by leveraging the multinerve and the Čech complex, removing local finiteness constraints. The central result, the Multinerve Theorem, shows that under $(n-|\\mathcal{F}|+1)$-connectedness of finite intersections, the homotopy type of the space matches the multinerve up to degree $n$ (with a surjection at $n+1$), via a natural zig-zag of weak equivalences and $(n+1)$-connected maps. The paper also develops partial nerves and complete CW- and poset-theoretic frameworks, leading to generalized crosscut results and variations on Quillen’s Poset Fiber Theorem, with broad applications in topological combinatorics. Overall, the approach unifies and extends nerve-type results across diverse settings, providing robust combinatorial models for homotopy types and new tools for poset topology.

Abstract

Given a locally finite cover of a simplicial complex by subcomplexes, Björner'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 Verdière, Ginot, and Goaoc. Our main tool is the Čech 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.

Variations on the Nerve Theorem

TL;DR

This work broadens the classical Nerve Theorem to covers of CW complexes and open covers of arbitrary spaces by leveraging the multinerve and the Čech complex, removing local finiteness constraints. The central result, the Multinerve Theorem, shows that under -connectedness of finite intersections, the homotopy type of the space matches the multinerve up to degree (with a surjection at ), via a natural zig-zag of weak equivalences and -connected maps. The paper also develops partial nerves and complete CW- and poset-theoretic frameworks, leading to generalized crosscut results and variations on Quillen’s Poset Fiber Theorem, with broad applications in topological combinatorics. Overall, the approach unifies and extends nerve-type results across diverse settings, providing robust combinatorial models for homotopy types and new tools for poset topology.

Abstract

Given a locally finite cover of a simplicial complex by subcomplexes, Björner'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 Verdière, Ginot, and Goaoc. Our main tool is the Čech 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.
Paper Structure (13 sections, 34 theorems, 83 equations, 1 figure)

This paper contains 13 sections, 34 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nerve constructions
  3. The multinerve
  4. Covered spaces and functoriality of the multinerve
  5. The Multinerve Theorem
  6. The Čech complex
  7. Multinerves of locally open covers
  8. CW neighborhoods
  9. Multinerves of CW covers
  10. Partial nerves
  11. Cutsets in partially ordered sets
  12. Detection of $n$--connected maps and variations on Quillen's Poset Fiber Theorem
  13. Nerves of simplicial covers

Key Result

Theorem 1.1

Let $\mathcal{V}$ be either an open cover of a locally path connected topological space $X$, or a cover of a CW complex $X$ by subcomplexes. Assume that for some $n\geqslant 0$ and each $\mathcal{F} \subseteq \mathcal{V}$ with cardinality $0< |\mathcal{F}|\leqslant n$, each path component of $\bigca

Figures (1)

  • Figure 1: A CW decomposition of the sphere union an equatorial disk

Theorems & Definitions (88)

  • Theorem 1.1
  • Lemma 1.2: Ebert--Randal-Williams
  • Proposition 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 78 more