Tangential smoothings of $k$-fold connected sum of complex projective spaces

Abstract

This paper determines tangential structure set of ${\#}_k\mathbb{C}{P}^n$, for $3\le n\le 7$, by analyzing their stable cohomotopy groups and $KO$-groups. As a consequence, it establishes the existence of manifolds with tangential homotopy type but not homeomorphic to ${\#}_k\mathbb{C}{P}^n$ for $n=4,5$.

15 pages