Homotopy representations of the unitary groups

Lubawski, Wojciech; Ziemiański, Krzysztof

doi:10.2140/agt.2016.16.1913

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Homotopy representations of the unitary groups

Wojciech Lubawski and Krzysztof Ziemiański

Algebraic & Geometric Topology 16 (2016) 1913–1951

DOI: 10.2140/agt.2016.16.1913

Abstract

Let $G$ be a compact connected Lie group and let $ξ, ν$ be complex vector bundles over the classifying space $B G$ . 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 $ξ | B N_{p}^{G}$ , where $N_{p}^{G}$ is a maximal $p$ –toral subgroup of $G$ , contains a subbundle isomorphic to $ν | B N_{p}^{G}$ . 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 $B U (n)$ . 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.

Keywords

homotopy representation, classifying space, unitary group

Mathematical Subject Classification 2010

Primary: 55R37

Secondary: 55S35

References

Publication

Received: 26 June 2014

Revised: 26 October 2015

Accepted: 3 November 2015

Published: 12 September 2016

Authors

Wojciech Lubawski
Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa Poland

Krzysztof Ziemiański
Faculty of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2, 02-097 02-097 Warszawa Poland

Volume 16, issue 4 (2016)