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In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 {\displaystyle n\geq 1}. That is,
and
Therefore X is a connected space, with one non-zero higher Betti number, namely, b n = 1 {\displaystyle b_{n}=1}. It does not follow that X is simply connected, only that its fundamental group is perfect.
A rational homology sphere is defined similarly but using homology with rational coefficients.
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