Higher topological cyclic homology and the Segal conjecture for tori (Q616907)

In the chromatic picture of homotopy theory, different theories catch information at different heights: cohomology at height \(0\), \(K\)-theory height \(1\), elliptic cohomology height \(2\), etc. While there are geometric approaches to these three, other higher chromatic theories, such as the Morava \(K\)-theories, are constructed purely homotopy theoretically. The authors construct in this paper a higher topological cyclic homology and propose it as a more geometric approach to higher chromatic information. This is motivated by the red-shift conjecture of Madsen and Rognes according to which the \(n\)th iterated algebraic \(K\)-theory of a ring spectrum is of chromatic height \(n\), and the fact that the cyclotomic trace from algebraic \(K\)-theory to topological cyclic homology is a relative equivalence -- with the latter significantly easier to compute. In brief the construction is as follows. Topological cyclic homology of a connective spectrum \(A\) is obtained from the fixed point data of topological Hochschild homology, a spectrum that is non-equivariantly equivalent to \(A \otimes S^1\), see [\textit{L. Hesselholt} and \textit{Ib Madsen}, Topology 36, No.~1, 29--101 (1997; Zbl 0866.55002)]. Height \(n\) topological cyclic homology is constructed in a similar way where the circle \(S^1\) is replaced by the \(n\)-dimensional torus. The main theorem of the paper is a description of relations between restriction, Frobenius, Verschiebung and differential maps, which for \(n = 1\) generalize results that proved powerful calculational tools in the hands of Hesselholt and Madsen. As an application of their theory the authors give a description for the cohomotopy type of the classifying space of tori (the Segal conjecture for the tori).