Editing Homology sphere (section)

{{Short description|Topological manifold whose homology coincides with that of a sphere}}
In [[algebraic topology]], a '''homology sphere''' is an ''n''-[[manifold]] ''X'' having the [[homology group]]s of an ''n''-[[sphere]], for some integer <math>n\ge 1</math>. That is,

:<math>H_0(X,\Z) = H_n(X,\Z) = \Z</math> 

and

:<math>H_i(X,\Z) = \{0\}</math> for all other ''i''.

Therefore ''X'' is a [[connected space]], with one non-zero higher [[Betti number]],  namely, <math>b_n=1</math>. It does not follow that ''X'' is [[simply connected]], only that its [[fundamental group]] is [[perfect group|perfect]] (see [[Hurewicz theorem]]).

A [[rational homology sphere]] is defined similarly but using homology with rational coefficients.