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1. Spectral Waldhausen categories, the S•-construction, and the Dennis trace

   Campbell, Jonathan A; Lind, John A; Malkiewich, Cary; Ponto, Kate; Zakharevich, Inna (January 2025, The Graduate journal of mathematics)

   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. 

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2. A Shadow Perspective on Equivariant Hochschild Homologies

   https://doi.org/10.1093/imrn/rnac250  

   Adamyk, Katharine; Gerhardt, Teena; Hess, Kathryn; Klang, Inbar; Kong, Hana Jia (September 2022, International Mathematics Research Notices)

   Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $$\mathsf V$$, as well as for small $$\mathsf V$$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $$C_n$$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $$K$$-theory to twisted topological Hochschild homology. 

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3. On the K-theory of division algebras over local fields

   https://doi.org/10.1007/s00222-019-00909-x  

   Hesselholt, Lars; Larsen, Michael; Lindenstrauss, Ayelet (July 2019, Inventiones mathematicae)

   Let K be a complete discrete valuation field with finite residue field of characteristic p, and let D be a central division algebra over K of finite index d. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers ℓ different from p and integers j≥1 , there exists a "reduced norm" isomorphism of ℓ-adic K-groups Nrd_{D/K}:K_j(D,Z_ℓ)→K_j(K,Z_ℓ) such that d⋅Nrd_{D/K} is equal to the norm homomorphism N_{D/K}. The purpose of this paper is to prove the analogous result for the p-adic K-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence Trd_{A/S}:THH(A|D,Z_p)→THH(S|K,Z_p) between two p-complete cyclotomic spectra associated with D and K, respectively. Interestingly, we show that if p divides d, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, d⋅Trd_{A/S} agrees with the trace Tr_{A/S}, although this is possible as maps of spectra with T-action 

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4. A Trace Map on Higher Scissors Congruence Groups

   https://doi.org/10.1093/imrn/rnae153  

   Bohmann, Anna_Marie; Gerhardt, Teena; Malkiewich, Cary; Merling, Mona; Zakharevich, Inna (August 2024, International Mathematics Research Notices)

   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. 

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5. Topological Cyclic Homology Via the Norm

   Angeltveit, Vigleik; Blumberg, Andrew J.; Gerhardt, Teena; Hill, Michael A.; Lawson, Tyler; Mandell, Michael A. (October 2018, Documenta mathematica)

   We describe a construction of the cyclotomic structure on topological Hochschild homology (THH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (TC) and TR-theory: one starting with a ring C_n-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory over A in terms of TR over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on TR and TC. 

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