**Background:**

In this final Millennium Problem of the month of October, we visit a problem that has actually already been solved, but with the idea that things can still be learned from it. The Poincaré conjecture arises from the field of topology and involves the

*3-sphere*, the hypersphere that bounds the 4-dimensional unit ball. The theorem says

*"Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere."*At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology. In particular, he was interested in what topological properties characterized a sphere. Working with mathematician Enrico Betti, this led to the creation of the Poincaré conjecture. The Poincaré conjecture would remain unproven for years, proving only to be a tricky problem to solve. Until 2002, when Russian mathematician Grigori Perelman solved the problem using a concept known as

*Ricci flow,*using equations similar to the Heat equation in fluid dynamics. In 2010 Perelman was awarded both the million-dollar cash prize from the Clay Mathematics Institute, and the Fields Medal. He declined both, stating that his contributions toward solving the problem were no greater than those who came before him.

**Your Task:**

Though the problem has already been solved, there's plenty to gain from looking at it in detail. Including proving the equivalent form of the conjecture, which states

*"If a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it."*For more information, visit en.wikipedia.org/wiki/Poincaré_conjecture.