On trivialities of Chern classes

TL;DR

This paper characterizes when certain suspended stunted projective spaces are C-trivial, meaning all complex vector bundles over them have trivial total Chern class, providing a complete classification for these spaces.

Contribution

It completely determines the C-triviality of suspensions of stunted real, complex, and quaternionic projective spaces, a problem previously not fully resolved.

Findings

Identifies conditions for C-triviality in these spaces

Provides a complete classification for suspensions of stunted projective spaces

Advances understanding of Chern classes in topological spaces

Abstract

A finite $CW$-complex $X$ is $C$-trivial if for every complex vector bundle $\xi$ over $X$, the total Chern class $c(\xi)=1$. In this note we completely determine when each of the following spaces are $C$-trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces and suspensions of stunted quaternionic projective spaces.