Title: On topological cyclic homology,
Abstract: Topological cyclic homology is a refinement of Connes--Tsygan's cyclic
homology which was introduced by Bökstedt--Hsiang--Madsen in 1993 as an
approximation to algebraic $K$-theory. There is a trace map from algebraic
$K$-theory to topological cyclic homology, and a theorem of
Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of
relative theories for nilpotent immersions, which gives a way for computing
$K$-theory in various situations. The construction of topological cyclic
homology is based on genuine equivariant homotopy theory, the use of explicit
point-set models, and the elaborate notion of a cyclotomic spectrum.
The goal of this paper is to revisit this theory using only
homotopy-invariant notions. In particular, we give a new construction of
topological cyclic homology. This is based on a new definition of the
$\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be
a spectrum $X$ with $S^1$-action (in the most naive sense) together with
$S^1$-equivariant maps $\varphi_p: X\to X^{tC_p}$ for all primes $p$. Here
$X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ is the Tate
construction. On bounded below spectra, we prove that this agrees with previous
definitions. As a consequence, we obtain a new and simple formula for
topological cyclic homology.
In order to construct the maps $\varphi_p: X\to X^{tC_p}$ in the example of
topological Hochschild homology we introduce and study Tate diagonals for
spectra and Frobenius homomorphisms of commutative ring spectra. In particular
we prove a version of the Segal conjecture for the Tate diagonals and relate
these Frobenius homomorphisms to power operations.