Homology manifold

In mathematics, a homology manifold (or generalized manifold) is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory.

Definition

A homology G-manifold (without boundary) of dimension n over an abelian group G of coefficients is a locally compact topological space X with finite G-cohomological dimension such that for any xX, 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 vanish at 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 n1 dimensional homology manifold (without boundary).

Examples

Properties

References

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