Trace and transfer maps in the algebraic \(K\)-theory of spaces
type: Item
instance of: scholarly article
title: Trace and transfer maps in the algebraic \(K\)-theory of spaces
zbMATH Open document ID: 1116.19001
author: Christopher L. Douglas
published in: \(K\)-Theory
publication date: 2007-02-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 55R12
zbMATH DE Number: 5125359
zbMATH Keywords: trace
zbMATH Keywords: transfer
zbMATH Keywords: fixed points
zbMATH Keywords: stable homotopy
MaRDI profile type: Publication
full work available at URL: s10977-005-3639-8
OpenAlex ID: W1967014855
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DOI: 10.1007/S10977-005-3639-8
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description: scientific article; zbMATH DE number 5125359