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 $\mathbb {Z}$) match those of the sphere. Specifically, a manifold $M$ is a homology sphere if its homology groups are as follows:

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

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 $SL(2,5)$) 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