Homology sphere
Topological manifold whose homology coincides with that of a sphere / From Wikipedia, the free encyclopedia
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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 . That is,
and
- for all other i.
Therefore X is a connected space, with one non-zero higher Betti number, namely, . 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.