Homology Sphere

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 = H_n(X,Z)

and

H_i(X,Z) = {0} for all other i.

Therefore X is a connected space, with one non-zero higher Betti number: b_n. 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.