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The bicategory of parameterized spectra has a remarkably rich structure. In particular, it is possible to take traces in this bicategory, which give classical invariants that count fixed points. We can also take equivariant traces, which give significant generalizations of the classical invariants that count periodic points. Unfortunately, the existence of these traces in general depends on technical statements about the bicategory that can be difficult to verify directly. In this paper, we demonstrate the effectiveness of two tools -- rigidity and deformable functors -- by using them establish the structure we need to take these equivariant traces and to construct periodic-point invariants in a formal way., Comment: 39 pages

We prove that the cut-and-paste K-theory of polyhedra, also called scissors congruence K-theory, is a homotopy orbit spectrum. More generally, we show that a broad class of algebraic K-theory constructions, based on covering families, all commute with the formation of homotopy orbits. In other words, "K-theory of coverings" always has a Farrell-Jones type isomorphism. Finally, we introduce a trace map for K-theory of coverings, and use it to prove the existence of some classes in the higher K-theory of polyhedra., Comment: 30 pages, 3 figures

We give an alternative treatment of the foundations of parametrized spectra, with an eye toward applications in fixed-point theory. We cover most of the central results from the book of May and Sigurdsson, sometimes with weaker hypotheses, and give a new construction of the bicategory $\mathcal Ex$ of parametrized spectra. We also give a careful account of coherence results at the level of homotopy categories. This is an expository work, akin to a set of lecture notes, encompassing and extending the more novel material in the paper "A convenient category of parametrized spectra." Our primary goal is to give careful and explicit explanations for the standard facts surrounding the bicategory of parametrized spectra and the Reidemeister trace. In particular, our proof of Costenoble-Waner duality gives a more explicit account of how the concrete index formulas of Dold are related to the abstract theory of bicategorical duality from May and Sigurdsson, Ponto, and Shulman., Comment: 183 pages including references and index, 5 figures. v3: small changes made to refer to the new condensed version "A convenient category of parametrized spectra." A short user's guide accompanies the text, see https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pdf