What is "homology sphere"

n. (context topology English) A manifold whose homology is the same as that of some sphere of the same dimension

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,

H(X,Z) = Z = H(X,Z)

and

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

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