Pseudoisotopies and the Bökstedt trace
type: Item
instance of: scholarly article
title: Pseudoisotopies and the Bökstedt trace
zbMATH Open document ID: 1213.19002
author: Bjørn Jahren
published in: Geometriae Dedicata
publication date: 2010-10-27T00:00:00Z
review text: The author studies the space \(\mathcal P(M)\) of smooth pseudo-isotopies of the compact smooth manifold \(M\). A motivation of the paper is the question whether \(\mathcal P(M)\) is disconnected whenever \(M\) is non-simply-connected. The author shows that \(\mathcal P(M)\) is disconnected if \(\dim M\geq 5\) and \(\pi_1(M)\) contains an element that is not conjugate to its square. It appears to be unknown if there exist nontrivial finitely presented groups where every element is conjugate to its square. (Such a group would be perfect, non-residually-finite, and centerless as the author observes.) It is easy to see, as the author notes, that \(W(M):=Wh_1^+(\pi_1(M);\,\mathbb Z/2\oplus\pi_2(M))\) is nontrivial if \(M\) is non-simply-connected. (We note that a presentation of \(W(M)\) and the geometry of its generators are discussed in the last section of \textit{A. E. Hatcher}'s paper [Ann. Math. (2) 102, 101--137 (1975; Zbl 0305.57009)]; a recent discussion of the history of corrections of the latter paper can be found in \textit{F. Waldhausen}, the author and \textit{J. Rognes} [preprint, http://folk.uio.no/rognes/papers/plmf.pdf].) In his 1973 book with Wagoner, Hatcher claimed that \(\pi_0(\mathcal P(M))\simeq W(M)\oplus Wh_2(\pi_1(M))\) if \(\dim M\geq 6\) (which would have settled the aforementioned motivating question). This assertion was corrected by \textit{K. Igusa} [Lect. Notes Math. 1946, 104--172 (1984; Zbl 0546.57015)], who used an extension of Hatcher--Wagoner methods to show that there is an exact sequence \[ K_3(\mathbb Z[\pi_1(M)])@>\chi>>W(M)@>\phi>>\pi_0(\mathcal P(M))\to Wh_2(\pi_1(M))\to 0 \] when \(\dim M\geq 7\) and gave examples showing that \(\chi\) is nonzero in general. In order to detect nontrivial elements in the image of \(\phi\), the author uses Bökstedt's trace map \(A(M)\to\Omega^\infty S^\infty(\Lambda M)\), where \(\Lambda M\) is the free loop space and \(A(M)\) Walhausen's \(A\)-theory. The author also re-establishes Igusa's exac…
Mathematics Subject Classification ID: 19J99
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 5806794
zbMATH Keywords: \(K\)-theory of spaces
zbMATH Keywords: pseudoisotopy theory
zbMATH Keywords: trace maps
zbMATH Keywords: Igusa exact sequence
reviewed by: Sergey A. Melikhov
MaRDI profile type: Publication
full work available at URL: s10711-009-9441-7
OpenAlex ID: W2128159465
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DOI: 10.1007/S10711-009-9441-7
description: scientific article; zbMATH DE number 5806794