nLab nerve theorem

Contents

This page is about a property of Cech nerves in homotopy theory. For the “nerve theorem” in category theory see at Segal conditions. For the “nerve theorem” for monads with arities see there.


Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The nerve theorem asserts that the homotopy type of a sufficiently nice topological space is encoded in the Čech nerve of a good open cover (as used in Čech homology).

This can be seen as a special case of some aspects of étale homotopy as the étale homotopy type of nice spaces coincides with the homotopy type of its Cech nerve.

Statement

Theorem

Let XX be a paracompact space and {U iX}\{U_i \to X\} a good open cover. Write C({U i})C(\{U_i\}) for the Cech nerve of this cover

C({U i})=( i,jU iU j iU i), C(\{U_i\}) = \left( \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} U_i \cap U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} U_i \right) \,,

(a simplicial space) and write

C˜({U i})=( i,j* i*) \tilde C(\{U_i\}) = \left( \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} * \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} * \right)

for the simplicial set obtained by replacing in C({U i})C(\{U_i\}) each direct summand space by the point. Let |C˜({U i})||\tilde C(\{U_i\})| be the geometric realization.

This is homotopy equivalent to XX.

The proof relies on the existence of partitions of unity.

This is usually attributed to (Borsuk 1948). It appears more explicitly as Weil 52, p. 141 McCord 67, Thm. 2, review in Hatcher, prop. 4G.3.

References

Original references:

  • Karol Borsuk, On the imbedding of systems of compacta in simplicial complexes , Fund. Math. 35, (1948) 217–234 (dml:213158)

  • Jean Leray, L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue, J. Math. Pures Appl. (9) 29 (1950), 1–139.

  • André Weil, §5. in: Sur les theoremes de de Rham, Comment. Math. Helv. 26 (1952), 119–145 (dml:139040)

  • Michael C. McCord, Homotopy type comparison of a space with complexes associated with its open covers, Proc. Amer. Math. Soc. 18 (1967), 705–708 (jstor:2035443)

  • Graeme Segal, §4 in: Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.

  • Armand Borel and Jean-Pierre Serre, Theorem 8.2.1. in: Corners and arithmetic groups,

    Comment. Math. Helv. 48 (1973), 436–491.

A version for hypercovers is discussed in

A review appears as corollary 4G.3 in the textbook

Some slightly stronger statements are discussed in

  • Anders Björner, Nerves, fibers and homotopy groups, Journal of combinatorial theory, series A, 102 (2003), 88-93

  • Andrzej Nagórko, Carrier and nerve theorems in the extension theory, Proc. Amer. Math. Soc. 135 (2007), 551-558. (web)

A nerve theorem for categories:

  • Kohei Tanaka, Cech complexes for covers of small categories, Homology, Homotopy and Applications 19(1), (2017), pp. 281-291. arXiv:1508.03688

Last revised on May 22, 2022 at 14:42:01. See the history of this page for a list of all contributions to it.