Homology sphere

Definition

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

$H_{i} (M; Z) = {\begin{matrix} Z & i = 0, n \\ 0 & i \neq 0, n \end{matrix}$

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

Note that we exclude the case $n = 0$ from consideration.

Examples

For any positive integer $n$ , the $n$ -sphere $S^{n}$ is a homology sphere.

For $n = 3$ , the Poincare homology sphere, obtained as the quotient of the 3-sphere by the binary icosahedral group (that is isomorphic to $S L (2, 5)$ ) 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
compact connected orientable manifold				\|FULL LIST, MORE INFO
For $n > 1$ , space with perfect fundamental group
For odd $n$ , space with Euler characteristic zero