Inertia Groups and Smooth Structures on Quaternionic Projective Spaces

TL;DR

This paper investigates the smooth structures on quaternionic projective spaces by computing inertia groups using stable homotopy theory, revealing trivial and non-trivial cases and their implications for manifold actions.

Contribution

It provides new computations of inertia groups and smooth structures on quaternionic projective spaces, extending understanding of their topology and symmetries.

Findings

Concordance inertia group is trivial in dimension 20.

Many high-dimensional cases have non-trivial concordance inertia groups.

Applications to 3-sphere actions and tangential homotopy structures.

Abstract

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia group and its analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.