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
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 $n>1$, space with perfect fundamental group
For odd $n$, space with Euler characteristic zero