- Homology manifold
In
mathematics , a homology manifold (or generalized manifold)is alocally compact topological space "X" that looks locally like atopological manifold from the point of view ofhomology theory .Definition
A homology "G"-manifold (without boundary) of dimension "n" over an abelian group "G" of coefficients is a locally compact topological space with finite "G"-
cohomological dimension such that for any "x"∈"X", the homology groups :H_p(X,X-x, G)are trivial unless "p"="n", in which case they are isomorphic to "G". Here "H" is some homology theory, usually singular homology. Homology manifolds are the same as homology Z-manifolds.More generally, one can define homology manifolds with boundary, by allowing the local homology groups to vanishat some points, which are of course called the boundary of the homology manifold. The boundary of an "n"-dimensional
first-countable homology manifold is an "n"−1 dimensional homology manifold (without boundary).Examples
*Any topological manifold is a homology manifold.
*An example of a homology manifold that is not a manifold is the suspension of ahomology sphere that is not a sphere.
*If "X"×"Y" is a topological manifolds, then "X" and "Y" are homology manifolds.References
*springer|id=H/h047800|title=Homology manifold|author=E. G. Sklyarenko
*W. J .R. Mitchell, " [http://links.jstor.org/sici?sici=0002-9939%28199010%29110%3A2%3C509%3ADTBOAH%3E2.0.CO%3B2-R Defining the boundary of a homology manifold] ",Proceedings of the American Mathematical Society , Vol. 110, No. 2. (Oct., 1990), pp. 509-513.
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