# Homology sphere

## Contents

## Definition

Suppose is a natural number. A **homology -sphere** is a -dimensional manifold whose homology groups (over the ring of integers ) match those of the sphere. Specifically, a manifold is a homology sphere if its homology groups are as follows:

Note that it is important to explicitly specify that the manifold is -dimensional, otherwise a cylinder over a sphere would satisfy the definition.

Note that we exclude the case from consideration.

## Examples

For any positive integer , the -sphere is a homology sphere.

For , the Poincare homology sphere, obtained as the quotient of the 3-sphere by the binary icosahedral group (that is isomorphic to ) is a homology sphere.

## Facts

### Complement of a point

The complement of any point in a homology sphere is an acyclic space. In particular, when the homology sphere is *not* a sphere (and specifically, its fundamental group is a nontrivial perfect group) then the complement of a point in it is acyclic but *not* weakly contractible.

### Suspension

- The suspension of any homology 3-sphere that is not homeomorphic to the 3-sphere is a homology manifold that is not a manifold.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

rational homology sphere | manifold whose homology groups over rationals match those of a sphere of the same dimension | |FULL LIST, MORE INFO | ||

compact connected orientable manifold | |FULL LIST, MORE INFO | |||

For , space with perfect fundamental group | ||||

For odd , space with Euler characteristic zero |