Power maps and epicyclic spaces (Q1336808)

Epicyclic spaces were introduced by Goodwillie in 1987 in an unpublished letter to Waldhausen. Goodwillie considered them as a framework for defining a topological variant of cyclic homology. (In the meantime of this theory has been developed by Goodwillie and Bökstedt-Hsiang- Madsen). An epicyclic space \(X\) is a cyclic space with additional operations \(p^k_n: X_n\to X_{k(n+ 1)-1}\) which are suggested by the \(k\)-th subdivision functor of the standard simplicial category \(\Delta\). Adding the duals of these operations to the cyclic category \(\Lambda\) we obtain the epicyclic category \(\widetilde {\Lambda}\). The authors determine the homotopy type of \(\widetilde {\Lambda}\) and study the structure of the topological realization of the underlying simplicial space of \(X\). The cyclic nerve \(N^{\text{cyc}} G\) of a topological group \(G\) extends in a canonical way to an epicyclic space. The additional structure extends the \(S^1\)-action on \(|N^{\text{cyc}} G|\) to an action of a semidirect product of \(S^1\) with \((\mathbb{N}, \cdot)\). The \(S^1\)-action on the free loop space \(\Lambda Y\) also extends to an action of this semidirect product. The authors show that the natural map \(|N^{\text{cyc}} G|\to \Lambda BG\) is equivariant with respect to these actions.