$C$-triviality of manifolds of low dimensions

TL;DR

This paper classifies low-dimensional closed smooth manifolds that are $C$-trivial, meaning all their complex vector bundles have trivial total Chern class, using homological methods and spectral sequences.

Contribution

It provides a complete homological classification of $C$-trivial manifolds below dimension 7, including orientable and non-orientable cases, employing the Atiyah-Hirzebruch spectral sequence.

Findings

Classified $C$-trivial manifolds in dimensions less than 7.

Established homological conditions for non-orientable $C$-trivial manifolds in dimension 7.

Used spectral sequence techniques to analyze Chern classes.

Abstract

A space $X$ is said to be $C$-trivial if the total Chern class $c(\alpha)$ equals $1$ for every complex vector bundle $\alpha$ over $X$. In this note we give a complete homological classification of $C$-trivial closed smooth manifolds of dimension $< 7$. In dimension $7$ we give a complete classification of orientable $C$-trivial manifolds and in the non-orientable case we give necessary homological conditions for the manifold to be $C$-trivial. Our main tool is the Atiyah-Hirzebruch spectral sequence and orders of its differentials.