High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories

Abstract

Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss'unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.