A. Suppose X is a finite complex. Is there a smooth, compact boundaryless manifold Mⁿ homotopy equivalent to X?

B. If yes, how many distinct such Mⁿ's are there, up to diffeomorphism?

C. Given two such manifolds Mⁿ and Nⁿ, and a homotopy equivalence f: M  N, is f homotopic to a diffeomorphism?

For question A one can write down a short list of necessary conditions for such an M to exist, the most obvious being Poincaré duality. The absolutely amazing fact is that these necessary conditions are almost sufficient, at least when X is simply-connected and n  5. This is made precise in a beautiful theorem of Browder. Questions B and C have equally beautiful answers due to Novikov. The main technique used is known as surgery, a paradoxical term since it involves killing selected homotopy groups of a manifold. Surgery was first introduced by Milnor, who says he got the idea from René Thom, and was subsequently generalized and applied by Kervaire-Milnor, Browder, Novikov and Wall.

We will spend much of the quarter studying in detail the landmark paper that started it all: Groups of Homotopy Spheres by Kervaire and Milnor (Annals of Math., 1963). This beautiful paper--one of my all-time favorites--answers question B in the case when X is the n-sphere, n > 4. In effect, it classifies the distinct differentiable structures on the sphere. Here "classify'' must be interpreted in the theological sense of reduction to an unsolved problem of homotopy theory. Nevertheless, much interesting, explicit information is obtained, and the Bernouilli numbers make a surprising appearance. In the last few weeks we will survey the Browder-Novikov theorems and their proofs (which rely heavily on the techniques of Kervaire-Milnor).

Along the way, we will meet a host of ideas and results of independent interest: characteristic classes, framed cobordism, stable homotopy groups of spheres, the Bott periodicity theorem, the Hirzebruch signature theorem.  Out of necessity, many of these background theorems will be presented without proof, and in a sense the course will also function as a broad survey of algebraic and differential topology. At the same time, it will give students an opportunity to see how more basic theorems such as Poincaré duality get used in practice--or as I like to put it, to see homology groups that live and breathe.

Prerequisites: All interested parties are welcome, but to get the most out of the course one should know the basics of smooth manifolds (544-5-6) and algebraic topology (564-5-6), and be familiar with vector bundles. It would help to already know something about characteristic classes, either from a topological source or from differential geometry, but I will not assume this.