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 Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology ($TC$) and $TR$-theory: one starting with a ring $C_n$-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum $A$. We describe spectral sequences computing the relative theory over $A$ 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 $TC$.

55P91, 19D55, 16E40

topological cyclic homology, multiplicative norm, cyclotomic spectrum, Adams operations

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Australian National University, Canberra, Australia

University of Texas, Austin, TX 78712, USA

Michigan State University, East Lansing, MI 48824, USA

University of California, Los Angeles, CA 90025, USA

University of Minnesota, Minneapolis, MN 55455, USA

Indiana University, Bloomington, IN 47405, USA