Topological Cyclic Homology Via the Norm

By Vigleik Angeltveit, Andrew Justin Blumberg, T. Gerhardt, M. Hill, Tyler Lawson, Michael A. Mandell. Documenta Mathematica, 2014

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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 theory $_ATR$ 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 $TC$.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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