The additivity and multiplicativity of fixed-point invariants

Mike Shulman (UC San Diego)

Wednesday 18th April, 2012 16:00-17:00 204

Abstract

The Lefschetz number of a continuous map is a classical invariant containing information about fixed points. It admits both an explicit geometrical description and an abstract description as a "categorical trace". The former is good for hands-on computations and for extracting concrete information, while the latter is good for proving general theorems. Such theorems include additivity and multiplicativity formulas for quotients and fiber bundles, which allow us to put together many easy calculations to achieve difficult ones. The usual abstract framework of categorical trace is adequate for many theorems, but it fails to imply a very good multiplicativity theorem for fiber bundles. I will explain how the more general framework of "bicategorical traces" introduced by Kate Ponto solves this problem, resulting in a very general multiplicativity formula that subsumes all others. It also yields a new, abstract proof of the additivity theorem for quotients. This is joint work with Kate Ponto.

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