The “fundamental theorem” for the algebraic K-theory of spaces: II—the canonical involution

Let Xmaps toA(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) image-construction. The main result is proved using this alternative discription.