Algebra and Geometry Seminar

USC, Kaprelian Hall room 414

The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes.

Building on work of Hu, we use Weiss calculus and a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. We compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. This is joint work with Hood Chatham and Yang Hu.