Exotic Smooth Structures on Products of Certain Smooth Manifolds with Spheres [HBNI Th282]

Abstract

The classification of smooth structures on manifolds is a central problem in differential topology, particularly for manifolds sharing the same underlying topological type. This thesis studies this problem for product manifolds of the form M x S^k, where 1 <= k <= 10 and M is a closed, oriented, connected smooth 4-manifold, a closed simply connected smooth 5- or 6-manifold, or a closed, oriented, 3-connected smooth 8-manifold.

We first study smooth structures on manifolds up to concordance. For a smooth manifold N of dimension at least 5, Kirby and Siebenmann identified the set of concordance classes of smooth structures C(N) with the set of homotopy classes of maps from N to Top/O. From this correspondence, we show that the concordance inertia group Ic(M x S^k) is determined by the stable top-cell attaching map of M. In particular, when the stable homotopy type of M is known, the group Ic(M x S^k) can be computed explicitly. By analyzing the stable cell structures of the above-mentioned manifolds M and using known computations of the stable homotopy groups of spheres, we compute Ic(M x S^k) for all 1 <= k <= 10 and classify smooth structures on M x S^k up to concordance for certain values of k.

The second part of the thesis addresses the classification of smooth structures up to diffeomorphism. Let S(N) denote the set of orientation-preserving diffeomorphism classes of smooth manifolds that are homeomorphic to a given smooth manifold N. The computation of C(N) plays a key role in this classification. The group of self-homeomorphisms Homeo(N) acts on C(N), and this action induces a one-to-one correspondence between S(N) and the orbit space C(N) / omega_0(Homeo(N)). Equivalently, there is a bijection:

C(N) / omega_0(Homeo(N)) <-> S(N)

Building on the computations of C(M x S^k) and using surgery theory, we analyze this action and determine the inertia group for the manifolds CP^2 x S^k for 4 <= k <= 6, CP^3 x S^k for 2 <= k <= 7, and HP^2 x S^1, where HP^2 denotes a projective plane-like smooth 8-dimensional manifold. Finally, we obtain a complete diffeomorphism classification of all smooth manifolds homeomorphic to these product spaces, including the case of CP^3 x S^1.