Homology sphere

Summary

A homology $n$-sphere is a closed $n$-manifold with the same integral homology groups as the $n$-sphere. The first example that is not a sphere is the Poincaré homology sphere, a $3$-manifold with a nontrivial fundamental group of order $120$. Homology spheres show that homology alone does not determine a manifold, and they were central in the history of the Poincaré conjecture.