Homology manifold

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This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

Suppose n is a nonnegative integer. A topological space X is said to be a homology manifold of dimension n if it satisfies both the following conditions:

H_i(X; X \setminus \{ x \}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & i = n \\ 0, & i \ne n\end{array} \right.

We can similarly define the concept of a homology G-manifold of dimension n for any abelian group: instead of looking at the homology groups, we look at the homology groups with coefficients in G. The usual definition of homology manifold therefore equals homology G-manifold.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
manifold |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally compact space |FULL LIST, MORE INFO