Homotopy representations of the unitary groups

Lubawski, Wojciech¹ ; Ziemiański, Krzysztof²

¹Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland
²Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, 02-097 Warszawa, Poland

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 1913-1951

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1913

Classification : 55R37, 55S35
Keywords: homotopy representation, classifying space, unitary group

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