Operations on A-theoretic nil-terms
Abstract
For a space X, we define Frobenius and Verschiebung operations on the nil-terms NA^{fd} (X) in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we obtain that the homotopy groups of NA^{fd} (X) are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung defines a Z[N_x]-module structure on the homotopy groups of NA^{fd} (X), with N_x the multiplicative monoid. We also we give a calculation of the homotopy groups of the nil-terms NA^{fd} (*) after p-completion for an odd prime p as Z_p[N_x]-modules up to dimension 4p-7. We obtain non-trivial groups only in dimension 2p-2, where it is finitely generated as a Z_p[N_x]-module, and in dimension 2p-1, where it is not finitely generated as a Z_p[N_x]-module.
Keywords
Cite
@article{arxiv.math/0702580,
title = {Operations on A-theoretic nil-terms},
author = {Joachim Grunewald and John R. Klein and Tibor Macko},
journal= {arXiv preprint arXiv:math/0702580},
year = {2014}
}
Comments
24 pages, modified version with a new proof of the main theorem