There is an integer, called the Lefschetz number, associated to each endomorphism of a closed smooth manifold. The Lefschetz fixed point theorem states that the Lefschetz number of an endomorphism with no fixed points is zero. Dold and Puppe gave a proof of this theorem using the trace in symmetric monoidal categories. They showed this perspective also gives some generalizations of the Lefschetz number to equivariant and fiberwise invariants. Unfortunately, the Lefschetz number of an endomorphism can also be zero when the endomorphism has fixed points and all endomorphisms homotopic to it have fixed points. The Lefschetz number admits a refinement, called the Reidemeister trace, that (with some hypotheses) is zero if and only if the endomorphism is homotopic to a fixed point free endomorphism. This gives a converse to the Lefschetz fixed point theorem. I will describe a generalization of Dold and Puppe's trace to a trace for bicategories. This new trace describes the Reidemeister trace. I will also indicate how this approach can be used to define equivariant generalizations of the Lefschetz number and Reidemeister trace. |
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