Alexander–Spanier cohomology

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

History

It was introduced by Template:Harvs for the special case of compact metric spaces, and by Template:Harvs 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

$df(x_{0},\ldots ,x_{p})=\sum _{i}(-1)^{i}f(x_{0},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{p}).$

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 Template:Harv and Alexander–Spanier cohomology with compact supports Template:Harv.

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.

References

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