homology sphere

A compact $n$ -manifold $M$ is called a homology sphere if its homology is that of the $n$ -sphere $S^{n}$ , i.e. $H_{0}(M;\\mathbb{Z})\\cong H_{n}(M;\\mathbb{Z})\\cong\\mathbb{Z}$ and is zero otherwise.

An application of the Hurewicz theorem and homological Whitehead theorem shows that any simply connected homology sphere is in fact homotopy equivalent to $S^{n}$ , and hence homeomorphic to $S^{n}$ for $n\eq 3$ , by the higher dimensional equivalent of the Poincaré conjecture.

The original version of the Poincaré conjecture stated that every 3 dimensional homology sphere was homeomorphic to $S^{3}$ , but Poincaré himself found a counter-example. There are, in fact, a number of interesting 3-dimensional homology spheres.

单词	HomologySphere
释义	homology sphere A compact $n$ -manifold $M$ is called a homology sphere if its homology is that of the $n$ -sphere $S^{n}$ , i.e. $H_{0}(M;\\mathbb{Z})\\cong H_{n}(M;\\mathbb{Z})\\cong\\mathbb{Z}$ and is zero otherwise. An application of the Hurewicz theorem and homological Whitehead theorem shows that any simply connected homology sphere is in fact homotopy equivalent to $S^{n}$ , and hence homeomorphic to $S^{n}$ for $n\eq 3$ , by the higher dimensional equivalent of the Poincaré conjecture. The original version of the Poincaré conjecture stated that every 3 dimensional homology sphere was homeomorphic to $S^{3}$ , but Poincaré himself found a counter-example. There are, in fact, a number of interesting 3-dimensional homology spheres.
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