A homology sphere referred to in mathematics one -dimensional manifold whose homology groups isomorphic to those of ordinary sphere are explicit or expressed an -dimensional manifold, for their singular homology groups
From the homology one can read off that a compact, connected manifold without boundary is. Generally, however, is not simply connected: Dividing the fundamental group by its commutator subgroup then one obtains a group that is isomorphic to the first homology group. This means you can only conclude from that the fundamental group of a perfect group, ie to its commutator subgroup is isomorphic, but need not be that trivial.
Historically, homology spheres were first considered in the -dimensional topology.
Poincaré believed initially that the homology ring should be sufficient to characterize the dimensional standard sphere uniquely. But he discovered a counter-example ( the so-called Poincaré homology sphere ) and then formulated the sharper Poincaré conjecture ( in which it is additionally required ), which was only about 100 years later proved by Perelman.
- Geometric topology