Topological classification of multiaxial $U(n)$-actions (with an appendix by Jared Bass)

Abstract

This paper begins the classification of topological actions on manifolds by compact, connected, Lie groups beyond the circle group. It treats multiaxial topological actions of unitary and symplectic groups without the dimension restrictions used in earlier works by M. Davis and W. C. Hsiang on differentiable actions. The general results are applied to give detailed calculations for topological actions homotopically modeled on standard multiaxial representation spheres. In the present topological setting, Schubert calculus of complex Grassmannians surprisingly enters in the calculations, yielding a profusion of “fake” representation spheres compared with the paucity in the previously studied smooth setting.