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.



Homotopy theory



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.



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.

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).


Some earlier references on the nerve theorem include

  • K. Borsuk, On the imbedding of systems of compacta in simplicial complexes , Fund. Math. 35, (1948) 217–234
  • 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, Sur les theoremes de de Rham, Comment. Math. Helv. 26 (1952), 119–145. (See §6.)

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

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

  • Armand Borel and Jean-Pierre Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. (See Theorem 8.2.1.)

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)

Revised on July 13, 2015 03:36:11 by Urs Schreiber (