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 $n\ne 3$, 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
Canonical name	HomologySphere
Date of creation	2013-03-22 13:56:10
Last modified on	2013-03-22 13:56:10
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	4
Author	bwebste (988)
Entry type	Definition
Classification	msc 57R60