# Topological cyclic homology via the norm

@article{Angeltveit2014TopologicalCH, title={Topological cyclic homology via the norm}, author={Vigleik Angeltveit and Andrew J. Blumberg and Teena Gerhardt and Matthew Hill and Tyler Lawson and Michael A. Mandell}, journal={arXiv: K-Theory and Homology}, year={2014} }

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 Bokstedt coherence machinery. We are able to define versions of topological cyclic homology ($TC$) and TR-theory relative to a cyclotomic commutative ring spectrum $A$. We describe spectral sequences computing this relative…

## 31 Citations

### Cyclotomic structure in the topological Hochschild homology of $DX$

- Mathematics
- 2015

Let $X$ be a finite CW complex, and let $DX$ be its dual in the category of spectra. We demonstrate that the Poincar\'e/Koszul duality between $THH(DX)$ and the free loop space $\Sigma^\infty_+ LX$…

### On the topological Hochschild homology of $DX$

- Mathematics
- 2015

We begin a systematic study of the topological Hochschild homology of the commutative ring spectrum $DX$, the dual of a finite CW-complex $X$. We prove that the "Atiyah duality" between $THH(DX)$ and…

### A spectrum-level Hodge filtration on topological Hochschild homology

- Mathematics
- 2014

We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum R, and more generally the factorization homology $$R \otimes X$$R⊗X for any…

### $K$-theory of endomorphisms, the $\mathit{TR}$-trace, and zeta functions

- Mathematics
- 2020

We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the $K$-theory of endomorphisms to topological restriction homology (TR). Along the…

### Computational tools for twisted topological Hochschild homology of equivariant spectra

- MathematicsTopology and its Applications
- 2022

### Comparing cyclotomic structures on different models for topological Hochschild homology

- MathematicsJournal of Topology
- 2019

The topological Hochschild homology THH(A) of an orthogonal ring spectrum A can be defined by evaluating the cyclic bar construction on A or by applying Bökstedt's original definition of THH to A .…

### Real topological Hochschild homology via the norm and Real Witt vectors

- Mathematics
- 2021

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order 2 to the orthogonal group O(2). From this perspective, we then prove a…

### On curves in K-theory and TR

- Mathematics
- 2021

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the∞-category of cyclotomic spectra with values in…

### Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

- MathematicsAnnales scientifiques de l'École Normale Supérieure
- 2022

We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in…

### On the geometric fixed-points of real topological cyclic homology

- Mathematics
- 2021

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over…

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We begin a systematic study of the topological Hochschild homology of the commutative ring spectrum $DX$, the dual of a finite CW-complex $X$. We prove that the "Atiyah duality" between $THH(DX)$ and…

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