Inertia groups of high-dimensional complex projective spaces

Abstract

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension $4n+1$, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is nontrivial in many cases. In complex dimension $9$, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.