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:

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

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.

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.