nLab
nerve theorem

This page is about a property of Čech nerves in homotopy theory. For the “nerve theorem” in category theory see at Segal conditions.


Context

Topology

Homotopy theory

Contents

Idea

The nerve theorem asserts that the homotopy type of a sufficiently nice topological space is encoded in the Cech nerve of a good cover.

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{\to}{\stackrel{\to}{\to}}\coprod_{i,j} U_i \cap U_j \stackrel{\to}{\to} \coprod_{i} U_i \right) \,,

(a simplicial space) and write

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

This is usually attributed to (Borsuk1948). The proof relies on the existence of partitions of unity (see for instance the review Hatcher, prop. 4G.2).

References

The nerve theorem is usually attributed to

  • K. Borsuk, On the imbedding of systems of compacta in simplicial complexes , Fund. Math 35, (1948) 217-234

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)

Revised on July 5, 2013 17:05:36 by Urs Schreiber (89.204.137.186)