Compactly supported homology: Difference between revisions
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is [[naturally isomorphic]] to the [[direct limit]] of the ''n''th relative homology groups of pairs (''Y'', ''B''), where ''Y'' varies over [[compact subspace]]s of ''X'' and ''B'' varies over compact subspaces of ''A''. |
is [[naturally isomorphic]] to the [[direct limit]] of the ''n''th relative homology groups of pairs (''Y'', ''B''), where ''Y'' varies over [[compact subspace]]s of ''X'' and ''B'' varies over compact subspaces of ''A''. |
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[[Singular homology]] is compactly supported, since each singular chain is a finite sum of [[simplices]], which are compactly supported. [[Strong homology]] is not compactly supported. |
[[Singular homology]] is compactly supported, since each singular chain is a finite sum of [[simplex|simplices]], which are compactly supported. [[Strong homology]] is not compactly supported. |
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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''. |
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''. |
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Revision as of 00:59, 2 March 2008
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.
Singular homology is compactly supported, since each singular chain is a finite sum of simplices, which are compactly supported. 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.
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