# 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 M. 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… Expand

#### 28 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 $\Sigma^\infty_+ LX$… Expand

Comparing cyclotomic structures on different models for topological Hochschild homology

- Mathematics
- 2017

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 Bokstedt's original definition of $THH$… Expand

On topological cyclic homology

- Mathematics
- 2017

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace… Expand

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… Expand

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… Expand

Computational tools for twisted topological Hochschild homology of equivariant spectra

- Mathematics
- 2020

Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on… Expand

Unwinding the relative Tate diagonal

- Mathematics
- 2020

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This… Expand

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

- Mathematics
- 2019

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… Expand

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… Expand

The Witt vectors for Green functors

- Mathematics
- Journal of Algebra
- 2019

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(-)$, and it describes the $E_2$ term… Expand

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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… Expand

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