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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. 

In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Cheeger–Chern–Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic K-theory of C. 

Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann–Roch theorems of [21, 14]. In this paper, we prove a broad generalization of these Riemann–Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to symmetric monoidal bicategories and from the Euler class to traces of non-identity maps. Our generalization also implies spectral Riemann–Roch theorems. We regard this result as an instantiation of a 2-dimensional generalized cobordism hypothesis. This perspective draws the result close to many others that generalize results about Euler characteristics and classes to bicategorical traces. 

The Mohave Rattlesnake (Crotalus scutulatus) is a highly venomous pitviper inhabiting the arid interior deserts, grasslands, and savannas of western North America. Currently two subspecies are recognized: the Northern Mohave Rattlesnake (C. s. scutulatus) ranging from southern California to the southern Central Mexican Plateau, and the Huamantla Rattlesnake (C. s. salvini) from the region of Tlaxcala, Veracruz, and Puebla in south-central Mexico. Although recent studies have demonstrated extensive geographic variation in venom composition and cryptic genetic diversity in this species, no modern studies have focused on geographic variation in morphology. Here we analyzed a series of qualitative, meristic, and morphometric traits from 347 specimens of C. scutulatus and show that this species is phenotypically cohesive without discrete subgroups, and that morphology follows a continuous cline in primarily color pattern and meristic traits across the major axis of its expansive distribution. Interpreted in the context of previously published molecular evidence, our morphological analyses suggest that multiple episodes of isolation and secondary contact among metapopulations during the Pleistocene were sufficient to produce distinctive genetic populations, which have since experienced gene flow to produce clinal variation in phenotypes without discrete or diagnosable distinctions among these original populations. For taxonomic purposes, we recommend that C. scutulatus be retained as a single species, although it is possible that C. s. salvini, which is morphologically the most distinctive population, could represent a peripheral isolate in the initial stages of speciation.