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Homotopy representations of the unitary groups

Wojciech Lubawski and Krzysztof Ziemiański

Algebraic & Geometric Topology 16 (2016) 1913–1951

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.

homotopy representation, classifying space, unitary group
Mathematical Subject Classification 2010
Primary: 55R37
Secondary: 55S35
Received: 26 June 2014
Revised: 26 October 2015
Accepted: 3 November 2015
Published: 12 September 2016
Wojciech Lubawski
Institute of Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-956 Warszawa
Krzysztof Ziemiański
Faculty of Mathematics, Informatics and Mechanics
University of Warsaw
Banacha 2, 02-097
02-097 Warszawa