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We give an explicit point-set construction of the Dennis trace map from the K-theory of endomorphisms K End(C) to topological Hochschild homology THH(C) for any spectral Waldhausen category C. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in C indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausen’s S•-construction for spectral Waldhausen categories, which is central to this account of the Dennis trace map. Our goals are both convenience and transparency—we provide all details except for a proof of the additivity theorem for THH, which is taken for granted—and the exposition is concerned not with originality of ideas, but rather aims to provide a useful resource for learning about the Dennis trace and its underlying machinery. 

We give an explicit point-set construction of the Dennis trace map from the K-theory of endomorphisms K End(C) to topological Hochschild homology THH(C) for any spectral Waldhausen category C. We describe the necessary technical foundations, most notably a well-behaved model for the spectral category of diagrams in C indexed by an ordinary category via the Moore end. This is applied to define a version of Waldhausen’s S•-construction for spectral Waldhausen categories, which is central to this account of the Dennis trace map. Our goals are both convenience and transparency—we provide all details except for a proof of the additivity theorem for THH, which is taken for granted—and the exposition is concerned not with originality of ideas, but rather aims to provide a useful resource for learning about the Dennis trace and its underlying machinery. 

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. 

We give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on π_0. 

Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors. 

Abstract We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $$n$$-periodic points from a self-map $$f$$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.