A compact $n$ -manifold $M$ is called a homology sphere if its homology is that of the $n$ -sphere $S^n$ , i.e. $H_0(M;\Z)\cong H_n(M;\Z)\cong\Z$ 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.