In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,
- H0(X,Z) = Z = Hn(X,Z)
and
- Hi(X,Z) = {0} for all other i.
Therefore X is a connected space, with one non-zero higher Betti number: bn. It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem).
A rational homology sphere is defined similarly but using homology with rational coefficients.
Read more about Homology Sphere: Poincaré Homology Sphere, Constructions and Examples, Invariants, Applications
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