Operations on the A-theoretic Nil-terms

For a space X, we define Frobenius and Verschiebung operations on the nil-terms NAfd ± (X) in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we obtain that the homotopy groups of NAfd ± (X) are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung defines a Z[N×]-module structure on the homotopy groups of NAfd ± (X), with N× the multiplicative monoid. We also give a calculation of the homotopy type of the nil-terms NAfd ± (∗) after p-completion for an odd prime p and their homotopy groups as Zp[N×]modules up to dimension 4p− 7. We obtain non-trivial groups only in dimension 2p−2, where it is finitely generated as a Zp[N×]-module, and in dimension 2p− 1, where it is not finitely generated as a Zp[N×]-module.