Traces in Algebraic K-theory and Topological Fixed Point Invariants

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One of the barriers to answering many math questions is that there is just too much information - too many examples and too many possibilities. There are many ways to attempt to manage this problem but many of them are based on not remembering some of the differences between our examples. One way to do this is to define a function, an invariant, that assigns some simpler information, say a number, to each of our examples. This by itself doesn't help make progress since there maybe no special pattern to the function and it may distinguish between too many examples. We can make this work much better by imposing conditions on these functions. One of the most important options is to ask that for a function that can determined by its values on smaller pieces. This perspective is very powerful and informs approaches to many topological invariants, but important invariants associated to fixed point theory have escaped its reach. The goal of this project is to rectify this and use the tools that prioritize this additivity to further develop fixed point invariants. The project also supports the PI's work with the socioeconomically diverse graduate student population at the University of Kentucky. The PI has worked to build a community where students can identify difficulties, feel comfortable asking questions, and learn to effectively advocate for themselves. The additivity of the Euler characteristic is one of the most important properties of this very important invariant. One way to capture this additivity is to observe that the Euler characteristic can be understood as the image of a class in algebraic K-theory. Classically, the generalizations of the Euler characteristic that relate to topological fixed point theory ignored the significance of additivity. The work in this project seeks to rectify this omission. Previous results suggest two to approaches to this goal. The first recognizes that the Reidemeister trace, a refinement of the Euler characteristic that gives a converse to the Lefschetz fixed point theorem, takes values in topological Hochschild homology. Topological Hochschild homology receives a map from topological restriction homology and it seems likely that the Reidemeister trace will lift through this map to a topologically meaningful class in topological restriction homology. An alternative approach is to start with the question of understanding K-theory of endomorphisms of modules over E-infinity ring spectra. From here the goal is to describe connections between the cyclotomic trace and trace in bicategories and symmetric monodial categories. In all of this work it also important to ground the results in topological meaning - to verify that classes constructed in these various groups are invariants associated to interesting questions. For example, these constructions should start by giving interesting invariants for periodic points and possibly extend to invariants for dynamical systems. This perspective also fits within a longer term goal of use fixed point theory as a test case for new methods of producing additive invariants that arise from wrong-way-maps or transfers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.