homology sphere

A compactPlanetmathPlanetmath n-manifold M is called a homology sphere if its homology is that of the n-sphere Sn, i.e. H0(M;)Hn(M;) and is zero otherwise.

An application of the Hurewicz theorem and homological Whitehead theoremMathworldPlanetmath shows that any simply connected homology sphere is in fact homotopy equivalent to Sn, and hence homeomorphic to Sn for n3, by the higher dimensional equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of the Poincaré conjecture.

The original version of the Poincaré conjecture stated that every 3 dimensional homology sphere was homeomorphic to S3, 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