A classifying space for commutativity in Lie groups

Adem, Alejandro¹ ; Gómez, José²

¹Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Room 121, Vancouver BC V6T 1Z2, Canada
²Departmento de Matemáticas, Universidad Nacional de Colombia, Calle 59A No 63 – 20, Bloque 43, Medellín, Colombia

Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 493-535

Résumé

In this article we consider a space BcomG assembled from commuting elements in a Lie group G first defined by Adem, Cohen and Torres-Giese. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that ℤ × BcomU is a loop space and define a notion of commutative K–theory for bundles over a finite complex X, which is isomorphic to [X, ℤ × BcomU]. We compute the rational cohomology of BcomG for G equal to any of the classical groups SU(r), U(q) and Sp(k), and exhibit the rational cohomologies of BcomU, Bcom SU and Bcom Sp as explicit polynomial rings.

Détail

DOI : 10.2140/agt.2015.15.493

Classification : 22E99, 55R35
Keywords: commuting elements, Lie groups, classifying spaces

Bibliographie

[1] A Adem, F R Cohen, Commuting elements and spaces of homomorphisms, Math. Ann. 338 (2007) 587

[2] A Adem, F R Cohen, J M Gómez, Stable splittings, spaces of representations and almost commuting elements in Lie groups, Math. Proc. Cambridge Philos. Soc. 149 (2010) 455

[3] A Adem, F R Cohen, E Torres-Giese, Commuting elements, simplicial spaces and filtrations of classifying spaces, Math. Proc. Cambridge Philos. Soc. 152 (2012) 91

[4] A Adem, J M Gómez, Equivariant K–theory of compact Lie group actions with maximal rank isotropy, J. Topol. 5 (2012) 431

[5] A Adem, J M Gómez, J A Lind, U Tillmann, Infinite loop spaces and nilpotent K–theory, in preparation

[6] R M Adin, Y Roichman, The flag major index and group actions on polynomial rings, European J. Combin. 22 (2001) 431

[7] E E Allen, The descent monomials and a basis for the diagonally symmetric polynomials, J. Algebraic Combin. 3 (1994) 5

[8] T J Baird, Cohomology of the space of commuting n–tuples in a compact Lie group, Algebr. Geom. Topol. 7 (2007) 737

[9] A Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Math. J. 13 (1961) 216

[10] T Bröcker, T Tom Dieck, Representations of compact Lie groups, 98, Springer (1985)

[11] J J Duistermaat, J A C Kolk, Lie groups, Springer (2000)

[12] J M Gómez, A basis for the diagonally signed-symmetric polynomials, Electron. J. Combin. 20 (2013)

[13] J M Gómez, A Pettet, J Souto, On the fundamental group of Hom(Zk,G), Math. Z. 271 (2012) 33

[14] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[15] M Hochster, J A Eagon, Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971) 1020

[16] J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and E∞ structures, Arch. Math. (Basel) 59 (1992) 115

[17] J A Lind, Diagram spaces, diagram spectra and spectra of units, Algebr. Geom. Topol. 13 (2013) 1857

[18] J P May, The geometry of iterated loop spaces, 271, Springer (1972)

[19] J P May, E∞ spaces, group completions, and permutative categories, from: "New developments in topology" (editor G Segal), London Math. Soc. Lecture Note Ser. 11, Cambridge Univ. Press (1974) 61

[20] J Mccleary, A user’s guide to spectral sequences, 58, Cambridge Univ. Press (2001)

[21] D Mcduff, G Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1975/76) 279

[22] E Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152

[23] A Pettet, J Souto, Commuting tuples in reductive groups and their maximal compact subgroups, Geom. Topol. 17 (2013) 2513

[24] D Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976) 167

[25] G Segal, Categories and cohomology theories, Topol. 13 (1974) 293

[26] J Souto, A remark on the homotopy equivalence of SUn and SLn(C), (2010)

[27] F Vaccarino, The ring of multisymmetric functions, Ann. Inst. Fourier (Grenoble) 55 (2005) 717