Trace and transfer maps in the algebraic \(K\)-theory of spaces
Let \(E\to B\) be a Hurewicz fibration with compact fibers, and consider the \(A\)-theory transfer \(B\to A(E)\), where \(A\) is Waldhausen's algebraic \(K\)-theory of spaces. Waldhausen's A-theory of spaces admits a natural splitting \(A(E)\simeq Q(E_+)\times Wh(E)\) where \(Q(E_+)\) signifies the infinite loop space associated to the suspension spectrum of \(E_+\) and \(Wh(E)\) is the smooth Whitehead spectrum. It is natural to ask how this splitting behaves with respect to the transfer. \textit{W. Dwyer, M. Weiss} and \textit{B. Williams} proved [Acta Math. 190, 1--104 (2003; Zbl 1077.19002)] under certain manifold conditions that the \(A\)-theory transfer \(B\to A(E)\) factors up to homotopy through the Becker-Gottlieb transfer \(B\to Q(E_+)\) and the splitting \(Q(E_+)\to A(E)\). In this paper the author shows (without the manifold assumption) that the composite \[ B\to A(E)\to A(*)\to Q(S^0) \] induced by \(E\to *\) and the trace \(A(*)\to Q(S^0)\) is weakly homotopic to \[ B\to Q(E_+)\to Q(S^0). \] This is achieved through a description of the trace \(A(*)\to Q(S^0)\) in terms of ENRs. One would have preferred to prove (as for the case described by Dwyer, Weiss and Williams) that the Becker-Gottlieb transfer is homotopic to the composite \(B\to A(E)\to Q(E_+)\), and the author expresses confidence that the methods used in the paper should lead to such a proof.
scientific article; zbMATH DE number 5125359
scientific article