Poincare Homology Sphere

The Poincare homology sphere is a specific three dimensional manifold that has the same homology groups as the ordinary three dimensional sphere, meaning it is indistinguishable from the sphere by that particular algebraic measure, yet it is not simply connected, since its fundamental group is a nontrivial group of order one hundred twenty known as the binary icosahedral group. It can be constructed as the quotient of the three dimensional sphere by the action of that group, or equivalently by a particular surgery performed on the trefoil knot. Henri Poincare constructed the space in 1904 as a counterexample to his own earlier, looser conjecture that any three dimensional manifold with the homology of a sphere must itself be a sphere, and the discovery led him to restate his conjecture in the stronger, simple connectivity form that became known as the Poincare conjecture, the problem later resolved by Grigori Perelman at the start of the twenty-first century.