Let X → A(X) denote the algebraic K-theory of spaces functor. In the first paper of this series, we showed A(X × S1) decomposes into a product of a copy of A(X), a delooped copy of A(X) and two homeomorphic nil terms. The primary goal of this paper is to determine how the "canonical involution" acts on this splitting. A consequence of the main result is that the involution acts so as to transpose the nil terms. From a technical point of view, however, our purpose will be to give another description of the involution on A(X) which arises as a (suitably modified) script capital L sign.-construction. The main result is proved using this alternative discription. © 2002 Elsevier Science B.V. All rights reserved.
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