In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.
History
It was introduced by James W. Alexander (1935) for the special case of compact metric spaces, and by Edwin H. Spanier (1948) for all topological spaces, based on a suggestion of Alexander D. Wallace.
Definition
If X is a topological space and G is an abelian group, then there is a cochain complex C whose p-th term \( {\displaystyle C^{p}} \) is the set of all functions from \( {\displaystyle X^{p+1}} \) to G with differential \( {\displaystyle d\colon C^{p}\to C^{p+1}} \) 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 \( C_{0} \) of functions that vanish in a neighborhood of the diagonal. The Alexander–Spanier cohomology groups \( {\displaystyle H^{p}(X,G)} \) are defined to be the cohomology groups of the quotient 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, James 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, Bibcode:1935PNAS...21..509A, doi:10.1073/pnas.21.8.509, ISSN 0027-8424, JSTOR 86360, PMC 1076641, PMID 16577676
Bredon, Glen E. (1997), Sheaf theory, Graduate Texts in Mathematics, 170 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0647-7, 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, doi:10.2307/2321782, 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", The Annals of Mathematics, Second Series, 49: 407–427, doi:10.2307/1969289, ISSN 0003-486X, JSTOR 1969289, MR 0024621
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