Compactly-supported homology
In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces
- (X, A)
is naturally isomorphic to the direct limit of the nth relative homology groups of pairs (Y, B), where Y varies over compact subspaces of X and B varies over compact subspaces of A.[1]
Singular homology is compactly supported, since each singular chain is a finite sum of simplices, which are compactly supported.[1] Strong homology is not compactly supported.
If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff pairs (X, A) with A closed in X, by defining that the homology of a Hausdorff pair (X, A) is the direct limit over pairs (Y, B), where Y, B are compact, Y is a subset of X, and B is a subset of A.
References
- 1 2 Kreck, Matthias (2010), Differential Algebraic Topology: From Stratifolds to Exotic Spheres, Graduate Studies in Mathematics, 110, American Mathematical Society, p. 95, ISBN 9780821848982.