## Algebraic & Geometric Topology

### Homotopy representations of the unitary groups

#### Abstract

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

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

Dates
Revised: 26 October 2015
Accepted: 3 November 2015
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895904

Digital Object Identifier
doi:10.2140/agt.2016.16.1913

Mathematical Reviews number (MathSciNet)
MR3546455

Zentralblatt MATH identifier
1351.55013

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

#### Citation

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. https://projecteuclid.org/euclid.agt/1511895904

#### References

• J,L Alperin, P Fong, Weights for symmetric and general linear groups, J. Algebra 131 (1990) 2–22 \goodbreak
• M Aschbacher, R Kessar, B Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series 391, Cambridge Univ. Press, MA (2011)
• A,K Bousfield, D,M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer, Berlin (1972)
• W Dwyer, A Zabrodsky, Maps between classifying spaces, from: “Algebraic topology”, (J Aguadé, R Kane, editors), Lecture Notes in Math. 1298, Springer, Berlin (1987) 106–119
• S Jackowski, J McClure, B Oliver, Homotopy classification of self-maps of $BG$ via $G$–actions I, Ann. of Math. 135 (1992) 183–226
• S Jackowski, J McClure, B Oliver, Homotopy classification of self-maps of $BG$ via $G$–actions II, Ann. of Math. 135 (1992) 227–270
• S Jackowski, J McClure, B Oliver, Maps between classifying spaces revisited, from: “The Čech centennial”, (M Cenkl, H Miller, editors), Contemp. Math. 181, Amer. Math. Soc., Providence, RI (1995) 263–298
• S Jackowski, B Oliver, Vector bundles over classifying spaces of compact Lie groups, Acta Math. 176 (1996) 109–143
• W Lubawski, K Ziemiański, Low dimensional homotopy representations of unitary groups In preparation
• D Notbohm, Maps between classifying spaces, Math. Z. 207 (1991) 153–168
• B Oliver, Higher limits via Steinberg representations, Comm. Algebra 22 (1994) 1381–1393
• B Oliver, $p$–stubborn subgroups of classical compact Lie groups, J. Pure Appl. Algebra 92 (1994) 55–78
• D Sullivan, Geometric topology: localization, periodicity, and Galois symmetry, Mimeographed notes, Massachusetts Institute of Technology (1970) \LaTeX version available at http://www.maths.ed.ac.uk/~ aar/books/gtop.pdf
• Z Wojtkowiak, On maps from ${\rm ho}\varinjlim F$ to ${\bf Z}$, from: “Algebraic topology”, (J Aguadé, R Kane, editors), Lecture Notes in Math. 1298, Springer, Berlin (1987) 227–236