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\section*{topological cyclic homology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesExamples}{Examples}\dotfill \pageref*{ReferencesExamples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Topological Hochschild homology} (resp. \textbf{topological cyclic homology}) (see the survey in (\hyperlink{May}{May})) is a refinement of [[Hochschild homology]]/[[cyclic homology]] from [[commutative rings]]/algebras to the [[higher algebra]] of [[ring spectra]]/[[E-∞ rings]]/[[E-∞ algebras]].

One motivation for their study comes from computational considerations, as in certain cases, these invariants are easier to compute than [[algebraic K-theory]], while there is a natural morphism of [[spectra]]

\begin{displaymath}
\itexarray{
     && \mathbf{TC}(R)
     \\
      & {}^{\mathllap{cyclotomic \atop trace}}\nearrow & \downarrow
     \\
     \mathbf{K}(R) &\underset{}{\longrightarrow}& \mathbf{THH}(R)
  }
\end{displaymath}
from the [[algebraic K-theory]] spectrum to the [[topological Hochschild homology]] spectrum, called the [[Dennis trace]] map, whose [[homotopy fiber|fiber]] is relatively well-understood. Since Hochschild homology spectra are naturally [[cyclotomic spectra]], this map factors through the topological cyclic homology spectrum via a map called the \emph{[[cyclotomic trace]]}, which acts much like a [[Chern character]] map for [[algebraic K-theory]].

The spectra $THH(R)$ and $TC(R)$ are typically easier to analyze than $K(R)$. Moreover, the difference between them and $K(R)$ is ``locally constant'' (\hyperlink{DundasGoodwillieMcCarthy13}{Dundas-Goodwillie-McCarthy13}) and often otherwise bounded in complexity. Accordingly, $THH$ and $TC$ are in practice computationally useful approximations to $K$.

There are various generalizations:

\begin{enumerate}%
\item Just as for basic Hochschild homology, there is \emph{higher topological Hochschild homology} (\hyperlink{CarlssonDouglasDundas08}{Carlsson-Douglas-Dundas 08}) given not just by [[derived loop spaces]] but by derived mapping spaces out of higher dimensional [[tori]].


\item Just as algebraic K-theory generalizes from [[E-∞ rings]] to [[stable ∞-categories]], so do $TC$ and the cyclotomic trace map (\hyperlink{BlumbergGepnerTabuada11}{Blumberg-Gepner-Tabuada 11})



\end{enumerate}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[cyclic homology]], [[dihedral homology]]


\item [[Chern character]]


\item [[norm cofibration]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The original references are

\begin{itemize}%
\item [[Marcel Bökstedt]], \emph{Topological Hochschild homology}, Bielefeld, 1985, 1988


\item [[Marcel Bökstedt]], W.C. Hsiang, [[Ib Madsen]], \emph{The cyclotomic trace and algebraic K-theory of spaces}, Invent. Math. \textbf{111} (1993), 463-539, \href{http://www.ams.org/mathscinet-getitem?mr=1202133}{MR94g:55011}, \href{http://dx.doi.org/10.1007/BF01231296}{doi}


\item [[Marcel Bökstedt]], [[Ib Madsen]], \emph{Topological cyclic homology of the integers}, $K$-theory (Strasbourg, 1992). Ast\'e{}risque \textbf{226} (1994), 7--8, 57--143.



\end{itemize}
and a further generalization is defined in

\begin{itemize}%
\item [[Bjørn Ian Dundas]], Randy McCarthy, \emph{Topological Hochschild homology of ring functors and exact categories}, J. Pure Appl. Algebra \textbf{109} (1996), no. 3, 231--294, \href{http://www.ams.org/mathscinet-getitem?mr=1388700}{MR97i:19001}, 

\end{itemize}
Higher topological Hochschild Homology is discussed in

\begin{itemize}%
\item [[Gunnar Carlsson]], [[Christopher Douglas]], [[Bjørn Ian Dundas]], \emph{Higher topological cyclic homology and the Segal conjecture for tori}, Adv. Math. \textbf{226} (2011), no. 2, 1823--1874, (\href{http://arxiv.org/abs/0803.2745}{arXiv:0803.2745}, \href{http://www.ams.org/mathscinet-getitem?mr=2737802}{MR2737802}, \href{http://dx.doi.org/10.1016/j.aim.2010.08.016}{doi})

\end{itemize}
A general abstract construction is in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Peter Scholze]], chapter 3 of \emph{On topological cyclic homology} (\href{https://arxiv.org/abs/1707.01799}{arXiv:1707.01799})

\end{itemize}
Review and exposition includes

\begin{itemize}%
\item [[Peter May]], \emph{Topological Hochschild and Cyclic Homology and Algebraic K-theory} (\href{http://www.math.uchicago.edu/~may/TALKS/THHTC.pdf}{pdf})


\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter IX of \emph{[[Rings, modules and algebras in stable homotopy theory]]}, AMS Mathematical Surveys and Monographs Volume 47 (1997) (\href{http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf}{pdf})


\item [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], \emph{The local structure of algebraic K-theory}, Springer 2013 (\href{http://math.mit.edu/~nrozen/juvitop/dundas.pdf}{pdf})


\item [[Teena Gerhardt]], \emph{Computations in algebraic K-theory}, talk at \href{http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html}{CUNY Workshop on differential cohomologies 2014} (\href{http://videostreaming.gc.cuny.edu/videos/video/1800/in/channel/55/}{video recording})


\item Ricardo Andrade, \emph{THH notes}, MIT juvitop seminar \href{http://math.mit.edu/~nrozen/juvitop/ra%20-%20juvitop.pdf}{pdf}, babytop seminar \href{http://math.mit.edu/~nrozen/juvitop/ra%20-%20babytop.pdf}{pdf}


\item Anatoly Preygel, \emph{Hochschild homology notes}, juvitop seminar, \href{http://math.mit.edu/~nrozen/juvitop/note_hh_talk.pdf}{pdf}


\item Thomas Geisser, \emph{Motivic Cohomology, K-Theory and Topological Cyclic Homology}, Handbook of K-theory II.1, \href{http://www.math.uiuc.edu/K-theory/handbook/1-193-234.pdf}{pdf}


\item Ib Madsen, \emph{Algebraic K-theory and traces}, \href{http://math.mit.edu/~nrozen/juvitop/madsen.pdf}{pdf}



\end{itemize}
Abstract characterization of the [[Dennis trace]] and cyclotomic trace is discussed in

\begin{itemize}%
\item [[Andrew Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], \emph{Uniqueness of the multiplicative cyclotomic trace}, Advances in Mathematics 260 (2014) 191-232 (\href{http://arxiv.org/abs/1103.3923}{arXiv:1103.3923})

\end{itemize}
An approach using only homotopy-invariant notions, which gives a construction of topological cyclic homology based on a new definition of the ∞-category of [[cyclotomic spectra]] is in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Peter Scholze]], \emph{On topological cyclic homology}, \href{https://arxiv.org/abs/1707.01799}{arXiv:1707.01799}

\end{itemize}
See also

\begin{itemize}%
\item T. Pirashvili, F. Waldhausen, \emph{Mac Lane homology and topological Hochschild homology}, J. Pure Appl. Algebra \textbf{82} (1992), 81-98, \href{http://www.ams.org/mathscinet/search/mathscinet-getitem?mr=520492}{MR96d:19005}, 


\item [[T. Pirashvili]], \emph{On the topological Hochschild homology of $\mathbf{Z}/p^k\mathbf{Z}$}, Comm. Algebra \textbf{23} (1995), no. 4, 1545--1549, \href{http://www.ams.org/mathscinet-getitem?mr=1317414}{MR97h:19007}, \href{http://dx.doi.org/10.1080/00927879508825293}{doi}


\item Z. Fiedorowicz, T. Pirashvili, R. Schw\"a{}nzl, R. Vogt, F. Waldhausen, \emph{Mac Lane homology and topological Hochschild homology}, Math. Ann. \textbf{303} (1995), no. 1, 149--164, \href{http://www.ams.org/mathscinet-getitem?mr=1348360}{MR97h:19007}, \href{http://dx.doi.org/10.1007/BF01460984}{doi}


\item [[Bjørn Ian Dundas]], \emph{Relative K-theory and topological cyclic homology}, Acta Math. \textbf{179} (1997), 223-242, (\href{http://link.springer.com/article/10.1007%2FBF02392744}{publisher})


\item Thomas Geisser, Lars Hesselhoft, \emph{Topological cyclic homology of schemes}, in: Algebraic $K$-theory (Seattle, WA, 1997), 41--87, Proc. Sympos. Pure Math. \textbf{67}, Amer. Math. Soc. 1999, \href{http://www.ams.org/mathscinet-getitem?mr=1743237}{MR2001g:19003}; \href{http://www.math.uiuc.edu/K-theory/0231/}{K-theory archive}


\item R. McCarthy, \emph{Relative algebraic K-theory and topological cyclic homology}, Acta Math. \textbf{179} (1997), 197-222.


\item J. McClure, R. Staffeldt, \emph{On the topological Hochschild homology of $b u$, I}, \href{http://math.mit.edu/~nrozen/juvitop/mcclure.pdf}{pdf}


\item Daniel Joseph Vera, \emph{Topological Hochschild homology of twisted group algebra}, MIT Ph. D. thesis 2006, \href{http://dspace.mit.edu/bitstream/handle/1721.1/34615/71329129.pdf?sequence=1}{pdf}


\item V. Angeltveit, A. Blumberg, T. Gerhardt, M. Hill, T. Lawson, M. Mandell, \emph{Topological cyclic homology via the norm} (\href{http://arxiv.org/abs/1401.5001}{arXiv:1401.5001})



\end{itemize}
\hypertarget{ReferencesExamples}{}\subsubsection*{{Examples}}\label{ReferencesExamples}

THH and TC specifically of [[KU|ku]] and [[KO|ko]] is discussed in

\begin{itemize}%
\item [[Christian Ausoni]], [[John Rognes]], \emph{Algebraic K-theory of topological K-theory}, Acta Mathematica March 2002, Volume 188, Issue 1, pp 1-39 (\href{http://www.math.uiuc.edu/K-theory/0405/}{KTheory 0405})


\item [[Vigleik Angeltveit]], [[Michael Hill]], [[Tyler Lawson]], \emph{Topological Hochschild homology of $\ell$ and $ko$} (\href{http://arxiv.org/abs/0710.4368}{arXiv:0710.4368})


\item [[Andrew Blumberg]], [[Michael Mandell]], \emph{Localization for $THH(ku)$ and the topological Hochschild and cyclic homology of Waldhausen categories} (\href{http://arxiv.org/abs/1111.4003}{arXiv:1111.4003})



\end{itemize}
and of [[tmf]] in

\begin{itemize}%
\item [[Bob Bruner]], [[John Rognes]], \emph{Topological Hochschild homology of topological modular forms} 2008 (\href{http://folk.uio.no/rognes/papers/ntnu08.pdf}{pdf})

\end{itemize}
[[!redirects topological Hochschild homology]]

[[!redirects THH]] [[!redirects TC]]



\end{document}