\(A^1\)-representability of hermitian \(K\)-theory and Witt groups
type: Item
instance of: scholarly article
title: \(A^1\)-representability of hermitian \(K\)-theory and Witt groups
zbMATH Open document ID: 1078.19004
author: Jens Hornbostel
published in: Topology
publication date: 2005-04-18T00:00:00Z
review text: The author proves various results about hermitian \(K\)-theory and Balmer's Witt groups. One of the main results is the representability of hermitian \(K\)-theory and Witt groups both in the unstable and the stable \(A^1\)-homotopy category. This result is similar to the representability result for algebraic \(K\)-theory proved by \textit{V. Voevodsky} [``\(A^1\)-homotopy theory'', Doc. Math., J DMV, Extra Vol. ICM, Berlin 1998, Vol. I, 579--604 (1998; Zbl 0907.19002)] and \textit{F. Morel} and \textit{V. Voevodsky} [``\(A^1\)-homotopy theory of schemes'', Publ. Math. IHÉS 90, 45--143 (1999; Zbl 0983.14007)]. In fact, the author proves a general representability result based only on the Mayer-Vietoris property of Nisnevich squares and homotopy invariance. To establish these properties for the algebraically defined Balmer Witt groups the author shows that in fact these groups arise naturally as homotopy groups of a topological space. As applications of the results the author computes hermitian \(K\)-theory of the projective line and discusses the relationship to Morel's conjecture about \(A^1\)-homotopy groups of spheres.
reviewed by: Manfred Kolster
Mathematics Subject Classification ID: 19G38
Mathematics Subject Classification ID: 19G12
zbMATH DE Number: 2158645
zbMATH Keywords: \(A^1\)-representability
zbMATH Keywords: hermitian \(K\)-theory
zbMATH Keywords: Witt groups
zbMATH Keywords: Bott periodicity
MaRDI profile type: Publication
full work available at URL: j.top.2004.10.004
OpenAlex ID: W1972427379
DOI: 10.1016/J.TOP.2004.10.004
description: scientific article; zbMATH DE number 2158645