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2016 Homotopy representations of the unitary groups
Wojciech Lubawski, Krzysztof Ziemiański
Algebr. Geom. Topol. 16(4): 1913-1951 (2016). DOI: 10.2140/agt.2016.16.1913


Let G be a compact connected Lie group and let ξ,ν be complex vector bundles over the classifying space BG. The problem we consider is whether ξ contains a subbundle which is isomorphic to ν. The necessary condition is that for every prime p, the restriction ξ|BN pG, where NpG is a maximal p–toral subgroup of G, contains a subbundle isomorphic to ν|BN pG. We provide a criterion when this condition is sufficient, expressed in terms of Λ –functors of Jackowski, McClure & Oliver, and we prove that this criterion applies for bundles ν which are induced by unstable Adams operations, in particular for the universal bundle over BU(n). Our result makes it possible to construct new examples of maps between classifying spaces of unitary groups. While proving the main result, we develop the obstruction theory for lifting maps from homotopy colimits along fibrations, which generalizes the result of Wojtkowiak.


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Wojciech Lubawski. Krzysztof Ziemiański. "Homotopy representations of the unitary groups." Algebr. Geom. Topol. 16 (4) 1913 - 1951, 2016.


Received: 26 June 2014; Revised: 26 October 2015; Accepted: 3 November 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 1351.55013
MathSciNet: MR3546455
Digital Object Identifier: 10.2140/agt.2016.16.1913

Primary: 55R37
Secondary: 55S35

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 4 • 2016
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