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Abstract Cut-and-paste $$K$$-theory has recently emerged as an important variant of higher algebraic $$K$$-theory. However, many of the powerful tools used to study classical higher algebraic $$K$$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $$K$$-theory. In this paper we address the particular case of the $$K$$-theory of polyhedra, also called scissors congruence $$K$$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $$K$$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $$K$$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $$K$$-theory. 

The topological Hochschild homology of a ring (or ring spectrum) R is an S1-spectrum, and the fixed points of THH(R) for subgroups C_n of S1 have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, drawing on the definition of equivariant Witt vectors by Blumberg, Gerhardt, Hill and Lawson, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.