TR and its relation to algebraic K-theory

The main object of study in this thesis is topological restriction homology (TR) which arises in the study of algebraic K-theory through the cyclotomic trace map to topological cyclic homology. ​Firstly, we construct TR in the formalism of Nikolaus—Scholze as a functor from cyclotomic spectra to spectra with Frobenius lifts by a suitable corepresentability formula inspired by work of Blumberg—Mandell. We prove that this construction agrees with the classical construction of TR as facilitated through genuine equivariant stable homotopy theory on bounded below cyclotomic spectra. As a consequence, we obtain a formula for TR in terms of Bloch’s spectrum of curves on algebraic K-theory generalizing work of Hesselholt and Betley—Schlichtkrull. ​Secondly, we introduce the notion of a polygonic spectrum which is a variant of the notion of a cyclotomic spectrum designed to capture the structure on topological Hochschild homology with coefficients. We define TR as a functor from polygonic spectra to spectra and prove that this construction agrees with the previously considered one for every cyclotomic spectrum regarded as a polygonic spectrum. We construct Verschiebung and Frobenius maps on TR of a polygonic spectrum and prove that these exhibit TR as the fixedpoints for a genuine action of the profinite integers on TR. Finally, inspired by the work of Kaledin, we construct the polygonic structure on topological Hochschild homology with coefficients by studying THH as a trace theory on a combinatorially defined category of cyclic graphs labelled by rings and bimodules.