An application of the Hurewicz theorem and homological Whitehead theorem shows that any simply connected homology sphere is in fact homotopy equivalent to , and hence homeomorphic to for , 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 , but Poincaré himself found a counter-example. There are, in fact, a number of interesting 3-dimensional homology spheres.
|Date of creation||2013-03-22 13:56:10|
|Last modified on||2013-03-22 13:56:10|
|Last modified by||bwebste (988)|