Journal of Topology Advance Access published online on December 25, 2007
Journal of Topology, doi:10.1112/jtopol/jtm012
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© 2007 London Mathematical Society
Operations on the A-Theoretic Nil-Terms
Mathematisches Institut Universität Münster Einsteinstrasse 62 Münster, D-48149 Germany grunewal{at}math.uni-muenster.de
Department of Mathematics Wayne State University Detroit, MI 48202 USA klein{at}math.waynexs.edu
Mathematisches Institut Universität Münster Einsteinstrasse 62 Münster, D-48149 Germany
Mathematical Institute SAS 
tef'anikova 49 Bratislava, SK-81473 Slovakia macko{at}math.uni-muenster.de
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Abstract |
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For a space X, we define Frobenius and Verschiebung operations on the nil-terms 
in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we show that the homotopy groups of 
are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung operation defines a 
-module structure on the homotopy groups of 
, with 
the multiplicative monoid.
We also give a calculation of the homotopy type of the nil-terms 
after p-completion for an odd prime p and their homotopy groups as 
-modules up to dimension 4p – 7. We obtain non-trivial groups only in dimension 2p – 2, where it is finitely generated as a 
-module, and in dimension 2p – 1, where it is not finitely generated as a 
-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.