New methods for enumerating non-stably trivial topological vector bundles

Given a finite CW complex X, a  (reduced) complex topological K-theory class h on X, and a positive integer r, it is classical that there are finitely many rank r unstable representatives for the K-theory class. However, actually enumerating such representatives is quite hard. When h=0 and $X=CP^n$, i.e., for stably trivial bundles on complex projective spaces, tools from Weiss calculus have been fruitfully applied to this enumeration by Hu, building on work of Weiss and Arone. In this talk, I will discuss new results that allow transfer of information from the case h=0 to the case h is non-zero. As an application, we give a complete enumeration of rank n vector bundles on $CP^{n+2}$ with K-theory class h, for any n and any h. This talk is based joint work-in-progress with Yang Hu.