About this book
This book is an exposition of the technique of surgery on simply-connected smooth manifolds. Systematic study of differentiable manifolds using these ideas was begun by Milnor  and Wallace  and developed extensively in the last ten years. It is now possible to give a reasonably complete theory of simply-connected manifolds of dimension ~ 5 using this approach and that is what I will try to begin here. The emphasis has been placed on stating and proving the general results necessary to apply this method in various contexts. In Chapter II, these results are stated, and then applications are given to characterizing the homotopy type of differentiable manifolds and classifying manifolds within a given homotopy type. This theory was first extensively developed in Kervaire and Milnor  in the case of homotopy spheres, globalized by S. P. Novikov  and the author  for closed 1-connected manifolds, and extended to the bounded case by Wall  and Golo . The thesis of Sullivan  reformed the theory in an elegant way in terms of classifying spaces.
Invariant Manifolds Mannigfaltigkeit character differentiable manifold duality form homotopy manifold proof proving theorem