Oscar Harr: Lefschetz–Nielsen fixed point theory in dualizable categories

Abstract:

In this talk, I report on ongoing joint work with Florian Riedel about (parametrized) Lefschetz–Nielsen theory. Broadly speaking, fixed point theory is concerned with the following question: how much can be said about the fixed point space \(X^f\) of a map \(f \colon X \longrightarrow X\) if we only have limited information about \(f\)? Concretely, this limited information may take the form of homotopical invariants associated with \(X\) (such as its Euler characteristic) or \(f\) (such as its Lefschetz trace), both of which have the advantage of being amenable to computation. A priori it is then surprising that anything at all can be deduced about \(X^f\) from this data, as the space of fixed points varies enormously as \(f\) varies over a homotopy class. Nevertheless, work by Nielsen, Reidemeister, and Wecken from the 1930's gives a homotopy-invariant lower bound on the number of fixed points of a map (in terms of the so-called Reidemeister trace, which refines the Lefschetz trace). This greatly strengthens the famous Lefschetz fixed point theorem for topological spaces. In my talk, I will present a unified approach to the Lefschetz fixed point theorem and (parametrized) Lefschetz–Nielsen theory, in which the whole theory emerges from an easy and conceptual calculation involving the derived category of sheaves on \(X\).