On topological cyclic homology

Abstract

Topological cyclic homology is a refinement of Connes–Tsygan’s cyclic homology which was introduced by Bokstedt–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 $\varphi_p : X \to X^{t C_p}$ for all primes $p$. Here, $X^{t C_p} = \mathrm{cofib}(\mathrm{Nm} : X^{h C_p} \to X^{h C_p})$ 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 $\varphi_p : X \to X^{t C_p}$ 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.