Topological cyclic homology via the norm

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 also able to define versions of topological cyclic homology (TC) and TR-theory relative to an arbitrary commutative ring spectrum A. We describe spectral sequences computing the 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.