Enumerating stably trivial vector bundles with higher real $K$-theory

Abstract

Given positive integers $r$ and $c$, let $\phi(r,c)$ denote the number of isomorphism classes of complex rank $r$ topological vector bundles on $\mathbb{CP}^{r+c}$ that are stably trivial. We compute the $p$-adic valuation of the number $\phi(r,c)$ for all pairs $r$ and $c$ such that $c \leq \operatorname{min}\{r,2p-3\}$. We also give some systematic lower bounds for $p$-divisibility of $\phi(r,c)$ when $c<2p^2-p-2$, and detect some nontrivial $p$-divisibility for larger $c$. As an additional application of our methods, we find new $p$-torsion in unstable homotopy groups of unitary groups.