The fundamental surgery theorem and the classification of manifolds

The purpose of this paper is to present a survey of some important results in the classification of differentiable manifolds. We begin with the Poincaré conjecture and its partial solution using the h-cobordism theorem. We review next the work of Kervaire and Milnor, concerned with the diffeomorphism classes of homotopy spheres. The surgery problem developed from their work, and we present its solution in the simply-connected case, by Browder. This solution amounts to the surgery invariant theorem, the fundamental surgery theorem and associated results. We end our discussion with the plumbing theorem, and several important classification theorems of Browder, Novikov and Wall.Science, Faculty ofMathematics, Department ofGraduat