Algebraic & Geometric Topology

Equivariant principal bundles and their classifying spaces

Wolfgang Lück and Bernardo Uribe

Full-text: Open access

Abstract

We consider Γ–equivariant principal G–bundles over proper Γ–CW–complexes with a prescribed family of local representations. We construct and analyze their classifying spaces for locally compact, second countable topological groups Γ and G with finite covering dimensions, where G is almost connected.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 4 (2014), 1925-1995.

Dates
Received: 17 April 2013
Revised: 7 January 2014
Accepted: 8 January 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715956

Digital Object Identifier
doi:10.2140/agt.2014.14.1925

Mathematical Reviews number (MathSciNet)
MR3331607

Zentralblatt MATH identifier
1307.55008

Subjects
Primary: 55R91: Equivariant fiber spaces and bundles [See also 19L47]
Secondary: 55P91: Equivariant homotopy theory [See also 19L47]

Keywords
equivariant principal bundle families of local representations classifying spaces

Citation

Lück, Wolfgang; Uribe, Bernardo. Equivariant principal bundles and their classifying spaces. Algebr. Geom. Topol. 14 (2014), no. 4, 1925--1995. doi:10.2140/agt.2014.14.1925. https://projecteuclid.org/euclid.agt/1513715956


Export citation

References

  • H Abels, A universal proper $G$–space, Math. Z. 159 (1978) 143–158
  • M F Atiyah, $K$–theory and reality, Quart. J. Math. Oxford Ser. 17 (1966) 367–386
  • M F Atiyah, Algebraic topology and operators in Hilbert space, from: “Lectures in Modern Analysis and Applications, I”, (C T Taam, editor), Springer, Berlin (1969) 101–121
  • M F Atiyah, G B Segal, Twisted $K$–theory, Ukr. Mat. Visn. 1 (2004) 287–330
  • T Barcenas, J Espinoza, M Joachim, B Uribe, Classification of twists in equivariant $K$–theory for proper and discrete group actions, Proc. Lond. Math. Soc. (2013)
  • G Birkhoff, A note on topological groups, Compositio Math. 3 (1936) 427–430
  • T tom Dieck, Transformation groups, Studies in Mathematics 8, de Gruyter, Berlin (1987)
  • S A Gaal, Linear analysis and representation theory, Grundlehren der Math. Wissenschaften 198, Springer, New York (1973)
  • I Hambleton, J-C Hausmann, Equivariant principal bundles over spheres and cohomogeneity one manifolds, Proc. London Math. Soc. 86 (2003) 250–272
  • P de la Harpe, Classical Banach–Lie algebras and Banach–Lie groups of operators in Hilbert space, Lecture Notes in Mathematics 285, Springer, Berlin (1972)
  • D Husemoller, Fibre bundles, McGraw-Hill Book Co., New York (1966)
  • S Illman, Existence and uniqueness of equivariant triangulations of smooth proper $G$–manifolds with some applications to equivariant Whitehead torsion, J. Reine Angew. Math. 524 (2000) 129–183
  • K Jänich, Vektorraumbündel und der Raum der Fredholm–Operatoren, Math. Ann. 161 (1965) 129–142
  • S Kakutani, Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. 12 (1936) 82–84
  • N Kitchloo, Dominant $K$–theory and integrable highest weight representations of Kac–Moody groups, Adv. Math. 221 (2009) 1191–1226
  • N H Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965) 19–30
  • S Lang, Differential manifolds, Addison-Wesley, Reading, MA (1972)
  • R K Lashof, Equivariant bundles, Illinois J. Math. 26 (1982) 257–271
  • R K Lashof, J P May, Generalized equivariant bundles, Bull. Soc. Math. Belg. Sér. A 38 (1986) 265–271
  • R K Lashof, J P May, G B Segal, Equivariant bundles with abelian structural group, from: “Proceedings of the Northwestern Homotopy Theory Conference”, (H R Miller, S B Priddy, editors), Contemp. Math. 19, Amer. Math. Soc. (1983) 167–176
  • D H Lee, T S Wu, On conjugacy of homomorphisms of topological groups, II, Illinois J. Math. 14 (1970) 409–413
  • W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Mathematics 1408, Springer, Berlin (1989)
  • W Lück, Survey on classifying spaces for families of subgroups, from: “Infinite groups: geometric, combinatorial and dynamical aspects”, (L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk, editors), Progr. Math. 248, Birkhäuser (2005) 269–322
  • J P May, Some remarks on equivariant bundles and classifying spaces, from: “International Conference on Homotopy Theory”, Astérisque 191, Soc. Math. France, Paris (1990) 7, 239–253
  • H Miyazaki, The paracompactness of $\mathit{CW}$–complexes, Tôhoku Math. J. 4 (1952) 309–313
  • P S Mostert, Local cross sections in locally compact groups, Proc. Amer. Math. Soc. 4 (1953) 645–649
  • J R Munkres, Topology: A first course, Prentice-Hall, Englewood Cliffs, N.J. (1975)
  • M Murayama, K Shimakawa, Universal equivariant bundles, Proc. Amer. Math. Soc. 123 (1995) 1289–1295
  • K-H Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2006) 291–468
  • R S Palais, On the existence of slices for actions of noncompact Lie groups, Ann. of Math. 73 (1961) 295–323
  • G B Segal, Cohomology of topological groups, from: “Symposia Mathematica, Vol. IV”, Academic Press, London (1970) 377–387
  • D J Simms, Topological aspects of the projective unitary group, Proc. Cambridge Philos. Soc. 68 (1970) 57–60
  • N E Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967) 133–152