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

TL;DR

This paper investigates the tangential structures of connected sums of complex projective spaces, revealing new manifolds with similar tangential properties but different topologies for certain dimensions.

Contribution

It determines the tangential structure set of connected sums of complex projective spaces for dimensions 3 to 7, introducing new manifolds with identical tangential types but distinct homeomorphism classes.

Findings

Identifies the tangential structure set for $ abla_k ext{CP}^n$ for 3 ≤ n ≤ 7.

Establishes existence of manifolds with tangential homotopy type but not homeomorphic to $ abla_k ext{CP}^n$ for n=4,5.

Abstract

This paper determines tangential structure set of $\#_k\mathbb{C}P^{n}$, for $3\leq n \leq 7$, 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 $n=4,5$.