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 x∈X, the homology groups 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 n−1 dimensional homology manifold (without boundary).

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