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.
For any positive integer , the -sphere 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.
- 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
|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|