2 0 Fe b 20 07 OPERATIONS ON THE A - THEORETIC NIL - TERMS

We define Frobenius and Verschiebung operations on the A-theoretic nil-terms NA ± (X) for a space X in three different ways. Two applications are included. Firstly, we obtain that the homotopy groups of NA ± (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 NA ± (X), with N× the multiplicative monoid. Further we give a calculation of the homotopy groups of the nil-terms NA ± (∗) after p-completion for an odd prime p 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.