Acyclic space: Difference between revisions
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A [[topological space]] is said to be '''acyclic''' if the homology groups in all dimensions are the same as those of a point, for any [[homology theory]]. Equivalently, it suffices to say that the singular homology groups are the same as those for a point. | A [[topological space]] is said to be '''acyclic''' if the homology groups in all dimensions are the same as those of a point, for any [[homology theory]]. Equivalently, it suffices to say that the singular homology groups are the same as those for a point. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Contractible space]] | |||
===Weaker properties=== | |||
* [[Space with finitely generated homology]] | |||
* [[Space with homology of finite type]] | |||
Revision as of 01:03, 27 October 2007
This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces
View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces
This is a variation of contractibility. View other variations of contractibility
Definition
A topological space is said to be acyclic if the homology groups in all dimensions are the same as those of a point, for any homology theory. Equivalently, it suffices to say that the singular homology groups are the same as those for a point.