Inertia groups of high dimensional complex projective spaces

TL;DR

This paper investigates the inertia groups of high-dimensional complex projective spaces, revealing their non-triviality in many cases and exploring implications for geometric structures in specific dimensions.

Contribution

It provides explicit computations of inertia groups using stable homotopy theory, especially in complex dimension 9, linking algebraic topology with geometric structures.

Findings

Inertia groups are isomorphic in complex projective spaces.

In complex dimension 4n+1, inertia groups relate to stable cohomotopy.

In complex dimension 9, inertia groups are often non-trivial.

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 non-trivial in many cases. In complex dimension 9, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.