This page is about a property of Čech nerves in homotopy theory. For the “nerve theorem” in category theory see at Segal conditions.
Paths and cylinders
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 be a paracompact space and a good open cover. Write for the Cech nerve of this cover
(a simplicial space) and write
for the simplicial set obtained by replacing in each direct summand space by the point. Let be the geometric realization.
This is homotopy equivalent to .
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).
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