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.
Connection to other cohomologies
- 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