# Homology manifold

From Topospaces

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

## Contents

## Definition

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

- is a locally compact space.
- For every point , the homology groups for the pair are as follows:

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