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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. That is, :''H''0(''X'',Z) = Z = ''H''''n''(''X'',Z) and :''H''''i''(''X'',Z) = for all other ''i''. Therefore ''X'' is a connected space, with one non-zero higher Betti number: ''bn''. 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. ==Poincaré homology sphere== The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Homology sphere」の詳細全文を読む スポンサード リンク
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