Alexander–Spanier cohomology

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

History[edit]

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[edit]

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

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.

Variants[edit]

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

Connection to other cohomologies[edit]

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

References[edit]