Algebraic & Geometric Topology

Homotopy representations of the unitary groups

Wojciech Lubawski and Krzysztof Ziemiański

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 4 (2016), 1913-1951.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R37: Maps between classifying spaces
Secondary: 55S35: Obstruction theory

homotopy representation classifying space unitary group


Lubawski, Wojciech; Ziemiański, Krzysztof. Homotopy representations of the unitary groups. Algebr. Geom. Topol. 16 (2016), no. 4, 1913--1951. doi:10.2140/agt.2016.16.1913.

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