$C$-triviality of manifolds of low dimensions

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.