The s-cobordism Theorem, the Sphere Embedding Theorem, and the Poincaré Conjecture

Abstract

The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture comprise three key consequences of the disc embedding theorem. The chapter begins by explaining in detail how to use the disc embedding theorem to prove the 5-dimensional s-cobordism theorem and the sphere embedding theorem. The sphere embedding theorem is the output from the disc embedding theorem that one wants in many situations. A version of the Poincaré conjecture is proven, specifically that every smooth homotopy 4-sphere is homeomorphic to the 4-sphere. All the results proved in this chapter are category losing; that is, they require smooth input but only produce homeomorphisms.