mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a homology manifold (or generalized manifold) is a locally compact topological space ''X'' that looks locally like a

topological manifold In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...

from the point of view of

homology theory In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...

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 In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomological ...

such that for any ''x''∈''X'', the

homology groups In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian group ...

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 In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...

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 a

homology sphere In algebraic topology, a homology sphere is an ''n''-manifold ''X'' having the homology groups of an ''n''-sphere, for some integer n\ge 1. That is, :H_0(X,\Z) = H_n(X,\Z) = \Z and :H_i(X,\Z) = \ for all other ''i''. Therefore ''X'' is a conne ...

that is not a sphere.

Properties

*If ''X''×''Y'' is a topological manifold, then ''X'' and ''Y'' are homology manifolds.

References

* * {{topology-stub Algebraic topology Generalized manifolds