Smooth Structures on a Fake Real Projective Space

TL;DR

This paper classifies all smooth structures on manifolds homotopy equivalent to the real projective 7-space, revealing exactly 28 and 56 distinct structures with the same PL and topological types, respectively.

Contribution

It provides a complete classification of smooth structures on manifolds homotopy equivalent to the real projective 7-space, including counts of distinct structures.

Findings

28 distinct smooth structures with same PL structure

56 distinct smooth structures with same topological structure

The group of smooth homotopy 7-spheres acts freely on these structures

Abstract

We show that the group of smooth homotopy $7$-spheres acts freely on the set of smooth manifold structures on a topological manifold $M$ which is homotopy equivalent to the real projective $7$-space. We classify, up to diffeomorphism, all closed manifolds homeomorphic to the real projective $7$-space. We also show that $M$ has, up to diffeomorphism, exactly $28$ distinct differentiable structures with the same underlying PL structure of $M$ and $56$ distinct differentiable structures with the same underlying topological structure of $M$.