Homology manifold

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:

• $X$ is a locally compact space.
• For every point $x\in X$, the homology groups for the pair $(X;X\setminus \{x\})$ are as follows:

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

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