Farrell-Jones spheres and inertia groups of complex projective spaces

TL;DR

This paper introduces Farrell-Jones spheres, a new class of homotopy spheres, and explores their role in constructing exotic negatively curved manifolds and their relation to inertia groups of complex projective spaces.

Contribution

It defines Farrell-Jones spheres, constructs exotic negatively curved manifolds using them, and analyzes their connection to inertia groups of complex projective spaces.

Findings

Every Hitchin sphere is a Farrell-Jones sphere.

Constructed negatively curved manifolds are homeomorphic but not diffeomorphic to complex hyperbolic manifolds.

Established a relationship between Farrell-Jones spheres and inertia groups of P^n.

Abstract

We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds $M^{2n}$, where $n=7$ or $8$, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C.S. Aravinda and F.T. Farrell. We show that every exotic sphere not bounding a spin manifold (Hitchin sphere) is a Farrell-Jones sphere. We also discuss the relationship between inertia groups of $\mathbb{C}\mathbb{P}^n$ and Farrell-Jones spheres.