#### Alladi Ramakrishnan Hall

#### Smooth Structures on the product of a manifold with a standard sphere

#### Ankur Sarkar

##### IMSc

*The study of exotic smooth structures on manifolds is one of the fundamental problems in topology. In particular, the classification of smooth structures on a given smooth manifold M is connected to the determination of a subgroup of the group of homotopy spheres, namely, the inertia group of M. More precisely, the inertia group of M is the collection of all exotic spheres whose connected sum with M is (oriented) diffeomorphic to M. In this talk, we compute the inertia group of the product of a manifold M with the standard k-sphere, where the dimension of M lies between 4 and 7 and k varies between 1 to 10. Using the stable homotopy type of M and the above computations of the inertia group, we classify all smooth manifolds homeomorphic to the product of M with the standard k-spheres upto*

concordance.

Done