# 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 $n\neq 3$, 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.

Title homology sphere HomologySphere 2013-03-22 13:56:10 2013-03-22 13:56:10 bwebste (988) bwebste (988) 4 bwebste (988) Definition msc 57R60