Farrell–Jones spheres and inertia groups of complex projective spaces

Abstract

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 2 n ${M^{2n}}$ , 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 ℂ ℙ n ${\mathbb {C}\mathbb {P}^n}$ and Farrell–Jones spheres.