Alexander–Spanier cohomology

In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.

History

It was introduced by J. W. Alexander (1935) for the special case of compact metric spaces, and by E. H. Spanier (1948) for all topological spaces, based on a suggestion of A. D. Wallace.

Definition

If X is a topological space and G is an abelian group, then there is a complex C whose pth term C^p is the set of all functions from X^p+1 to G with differential d given by

$df(x_0,\ldots,x_p)= \sum_i(-1)^if(x_0,\ldots,x_{i-1},x_{i+1},\ldots,x_p).$

It has a subcomplex C₀ of functions that vanish in a neighborhood of the diagonal. The Alexander–Spanier cohomology groups H^p(X,G) are defined to be the cohomology groups of the complex C/C₀.

Variants

It is also possible to define Alexander–Spanier homology (Massey 1978) and Alexander–Spanier cohomology with compact supports (Bredon 1997).

Connection to other cohomologies

The Alexander–Spanier cohomology groups coincide with Čech cohomology groups for compact Hausdorff spaces, and coincide with singular cohomology groups for locally finite complexes.