Lectures on Groups of Homotopy Spheres

Kervaire and Milnor's germinal paper [15], in which they used the newly-discovered techniques of surgery to begin the classification of smooth closed manifolds homotopy equivalent to a sphere (homotopy-spheres), was intended to be the first of two papers in which this classification would be essentially completed (in dimensions > 5). Unfortunately , the second part never appeared. As a result, in order to extract this classification from the published literature it is necessary to submerge oneself in more far-ranging and complicated works (e.g. [7], [16], [30]), which cannot help but obscure the beautiful ideas contained in the more direct earlier work of Kervaire and Milnor. This is especially true for the student who is encountering the subject for the first time. prepared mimeographed notes from these lectures, with some extra background material, which have been available from Brandeis University. The present article is almost identical with these notes. I hope it will serve to fill a pedagogical gap in the literature. The reader is assumed to be familiar with [15], [20]. In these papers, Kervaire-Milnor define the group en of h-cobordism classes of homotopy n-spheres and the subgroup bP n+l defined by homotopy spheres which bound parallelizable manifolds. The goal is to compute bP n+l and en/bpn+ i Section 1 reviews some well known results on vector bundles over spheres and the homotopy of the classical groups, as well as some theorems of Whitney on embeddings and immersions. Since a homotopy n-sphere ~n is h-cobordant to S n (the n-sphere with its standard differential structure) iff ~n bounds a contractible manifold, in order to calculate bP n+l we are interested in finding and realizing "obstructions" to surgering parallelizable manifolds into contractible