# 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 *p*th term *C*^{p} is the set of all functions from *X*^{p+1} to *G* with differential *d* given by

It has a subcomplex *C*_{0} 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*/*C*_{0}.

### 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.

## References

- Alexander, J. W. (1935), "On the Chains of a Complex and Their Duals",
*Proceedings of the National Academy of Sciences of the United States of America*, National Academy of Sciences,**21**(8): 509–511, doi:10.1073/pnas.21.8.509, ISSN 0027-8424, JSTOR 86360 - Bredon, Glen E. (1997),
*Sheaf theory*, Graduate Texts in Mathematics,**170**(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94905-5, MR 1481706 - Massey, William S. (1978), "How to give an exposition of the Čech-Alexander-Spanier type homology theory",
*The American Mathematical Monthly*,**85**(2): 75–83, ISSN 0002-9890, JSTOR 2321782, MR 0488017 - Massey, William S. (1978),
*Homology and cohomology theory. An approach based on Alexander-Spanier cochains.*, Monographs and Textbooks in Pure and Applied Mathematics,**46**, New York: Marcel Dekker Inc., ISBN 978-0-8247-6662-7, MR 0488016 - Spanier, Edwin H. (1948), "Cohomology theory for general spaces",
*Annals of Mathematics. Second Series*,**49**: 407–427, ISSN 0003-486X, JSTOR 1969289, MR 0024621