Show that a closed simply-connected 3-manifold is homotopy equivalent to S3. [Use Poincar\'e duality, and also the fact that closed manifolds are homotopy equivalent to CW complexes, from Corollary A.12 in the Appendix. The stronger statement that a closed simply-connected 3-manifold is homeomorphic to S3 is the Poincar\'e conjecture, finally proved by Perelman. The higher-dimensional analog, that a closed n-manifold homotopy equivalent to Sn is homeomorphic to Sn, had been proved earlier for all n≥4.]