Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer ${\displaystyle n\geq 1}$. That is,

${\displaystyle H_{0}(X,\mathbb {Z} )=H_{n}(X,\mathbb {Z} )=\mathbb {Z} }$

and

${\displaystyle H_{i}(X,\mathbb {Z} )=\{0\}}$ for all other i.

Therefore X is a connected space, with one non-zero higher Betti number, namely, ${\displaystyle b_{n}=1}$. 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.