Alexander–Spanier cohomology

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

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.

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

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.

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.

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