Alexander–Spanier cohomology

In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces, 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. It is also possible to define Alexander–Spanier homology (Massey 1978) and Alexander–Spanier cohomology with compact supports (Bredon 1997).

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

Definition

If X is a topological space and G is an abelian group, then there is a complex C whose pth term Cp is the set of all functions from Xp+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 Hp(X,G) are defined to be the cohomology groups of the complex C/C0.