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Journal of Topology Advance Access originally published online on December 25, 2007
Journal of Topology 2008 1(2):317-341; doi:10.1112/jtopol/jtm012
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© 2007 London Mathematical Society

Operations on the A-theoretic nil-terms

Joachim Grunewald

Mathematisches Institut, Universität Münster, Einsteinstrasse 62, Münster, D-48149, Germany grunewal@math.uni-muenster.de

John R. Klein

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA klein@math.wayne.edu

Tibor Macko

Mathematisches Institut, Universität Münster, Einsteinstrasse 62, Münster, D-48149, Germany
Mathematical Institute SAS, Stefánikova 49, Bratislava, SK-81473, Slovakia macko@math.uni-muenster.de

For a space X, we define Frobenius and Verschiebung operations on the nil-terms Formula in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we show that the homotopy groups of Formula are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung operation defines a Formula -module structure on the homotopy groups of Formula , with Formula the multiplicative monoid.

We also give a calculation of the homotopy type of the nil-terms Formula after p-completion for an odd prime p and their homotopy groups as Formula -modules up to dimension 4p – 7. We obtain non-trivial groups only in dimension 2p – 2, where it is finitely generated as a Formula -module, and in dimension 2p – 1, where it is not finitely generated as a Formula -module.

Received April 12, 2007.

2000 Mathematics Subject Classification 19D10, 19D35, 19D55, 55P42, 55P91

The first and the third author were supported by SFB 478 Geometrische Strukturen in der Mathematik, Münster and the second author was partially supported by the National Science Foundation.



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