Let Θm (m ≥ 5) be the group of diffeomorphism classes of smooth homotopy m-spheres. A possible way to change smooth structure on a smooth manifold Mm , without changing its homeomorphism type, is to take its connected sum Mm#Σm with a smooth homotopy sphere Σm . This induces a group action of Θm on the set of smooth structures on the topological manifold M. The collection of smooth homotopy spheres Σm which admit a diffeomorphism Mm#Σm → Mm forms a subgroup I(M) of Θm, called the inertia group of Mm . Analogous to the inertia group, for a manifold M one may define the homotopy inertia group Ih(M) and the concordance inertia group Ic(M). Ih(M) (respectively Ic(M)) consists of those Σ ∈ I(M) for which the diffeomorphism Mm#Σm → Mm is homotopic (respectively concordant) to the identity.
Computations in the inertia group are known for certain manifolds. However, there is no systematic approach for computing inertia groups in general, and many problems are still open. This talk deals with computations in the inertia groups of complex projective spaces, and associated computations for smooth structures.
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. By using the triviality and non-triviality of these inertia groups, we give two applications. As a first application, we classify all smooth manifolds homeomorphic to the complex projective n-space CPn up to isotopy for n ≤ 8. As a second application, we give examples of two inequivalent smooth structures of CP9 such that one admits a metric of non negative scalar curvature and the other does not.