On \([X,U(n)]\) when \(\dim X\) is \(2n\)
type: Item
instance of: scholarly article
title: On \([X,U(n)]\) when \(\dim X\) is \(2n\)
zbMATH Open document ID: 1070.55007
author: Hiroaki Hamanaka
author: Akira Kono
published in: Journal of Mathematics of Kyoto University
publication date: 2004-12-29T00:00:00Z
review text: For a CW complex \(X\) and the topological group \(U(n)\) the symbol \(U_n(X)\) denotes the group \([X,U(n)]\) of homotopy classes of maps of \(X\) into \(U(n)\). If dim\(X < 2n\), then \(U_n(X)\) equals \(\tilde{K}^1(X)\) and one may say that \(U_n(X)\) is ``the unstable \(\tilde{K}^1\)-theory''. The authors investigate, and obtain several results, the case when \(dim X = 2n\), in which case \(U_n(X)\) may be non-abelian. If \(\dim X \leq 2n\), they find a homomorphism \(\Theta: \tilde{K}^0(X) \to H^{2n}(X;\mathbb Z)\), such that the following exact sequence holds\newline \[ \tilde{K}^0(X) \to H^{2n}(X;\mathbb Z) \to U_n(X) \to \tilde{K}^1(X) \to 0 \] and it is natural. These results are based on investigations of the Coker\(\Theta = N_n(X)\). Among the results obtained we mention the following that has a simple formulation: If in addition \(X\) is finite, then \(N_n(X)\) is a finite Abelian group and the order of any element in \(N_n(X)\) divides \(n!\).
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 2124040
zbMATH Keywords: unitary group
zbMATH Keywords: CW complex
zbMATH Keywords: set of homotopy classes
zbMATH Keywords: K-theory
zbMATH Keywords: cohomology
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283730
OpenAlex ID: W1487388137
DOI: 10.1215/KJM/1250283730
description: scientific article; zbMATH DE number 2124040