mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a homology manifold (or generalized manifold) is a

locally compact topological space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...

''X'' that looks locally like a

topological manifold In topology, a branch of mathematics, 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 mathe ...

from the point of view of

homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

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, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

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 Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chromo ...

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