Operations on the A‐theoretic nil‐terms

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