Cobordism and Exotic Spheres

Preface Jules Henri Poincaré may rightly be considered the father of modern topology (Leonhard Euler and the Königsberg bridges notwithstanding). It is fitting, therefore, that many of the questions explored in this thesis originated with Poincaré. Poincaré's famous conjecture – which has driven so much of twentieth-century topology – arose, in fact, as the successor of an earlier conjecture. Poincaré originally conjectured that any 3-manifold with the homology of S 3 is homeomorphic to S 3. But he soon found a counterexample, known today as the Poincaré homology sphere (obtained from S 3 by surgery along the right trefoil with framing 1). As a result, Poincaré re-formulated his conjecture into the now-famous statement that any compact, connected, oriented 3-manifold with trivial fundamental group is homeomorphic to S 3. Although yet unresolved in dimension three, there is a natural generalization of Poincaré's conjecture stating that any compact, oriented n-manifold with the homotopy type of S n , n ≥ 4, is homeomorphic to S n. Stephen Smale resolved the Generalized Poincaré Conjecture affirmatively for n ≥ 5 in 1961. Finally, in 1982, Michael Freedman provided a proof in dimension four. Smale approached the Generalized Poincaré Conjecture by investigating the properties of mani-folds equivalent up to cobordism. As we shall see, cobordism theory is properly seen as a generalization of singular homology theory. It is ironic, therefore, in light of Poincaré's original homology conjecture, that the Generalized Poincaré Conjecture was resolved via cobordism theory. As it turns out, however, cobordism theory can address questions even more subtle than the Generalized Poincaré Conjecture. In fact, cobordism and surgery may be used together to distinguish between smooth n-manifolds which are homeomorphic but not diffeomorphic to S n. Such " exotic spheres " (or exotic differential structures on any manifold), were presumed nonexistent until John Milnor constructed the first one in 1956. The systematic characterization of exotic spheres, completed in 1963 by Milnor and Kervaire, relies heavily on cobordism, surgery, and the Hirzebruch signature formula. It is remarkable, in retrospect, that the cobordism tools first used by Smale to approach Poincaré's questions are also useful in the more subtle setting of exotic differential structures.