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Commuting elements, simplicial spaces and filtrations of classifying spaces

Published online by Cambridge University Press:  05 September 2011

ALEJANDRO ADEM
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver BC V6T1Z2, Canada. e-mail: [email protected]
FREDERICK R. COHEN
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A. e-mail: [email protected]
ENRIQUE TORRES GIESE
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A. e-mail: [email protected]

Abstract

Let G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that provide a filtration of the classifying space BG. In particular this setting gives rise to a single space constructed out of all the spaces of ordered commuting n–tuples of elements in G. Basic properties of these constructions are discussed, including the homotopy type and cohomology when the group G is either a finite group or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit–Thompson. The techniques here also yield a counting formula for the cardinality of Hom(π, G) where π is any descending central series quotient of a finitely generated free group. Another application is the determination of the structure of the spaces B(2, G) obtained from commuting n-tuples in G for finite groups such that the centralizer of every non–central element is abelian (known as transitively commutative groups), which played a key role in work by Suzuki on the structure of finite simple groups.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Adem, A.Cohomological exponents of ZG-lattices. J. Pure Appl. Algebra 58, no. 1 (1989), 15.CrossRefGoogle Scholar
[2]Adem, A., Bendersky, M., Bahri, A., Cohen, F. R. and Gitler, S.On decomposing suspensions of simplicial spaces. Bol. Soc. Mat. Mex. (3) 15 (2009), 91102.Google Scholar
[3]Adem, A. and Cohen, F.Commuting elements and spaces of homomorphisms. Math. Ann. 338 (2007), 587626.CrossRefGoogle Scholar
[4]Adem, A., Cohen, F. and Gómez, J.Stable spittings, spaces of representations and almost commuting elements in Lie groups. Math. Proc. Camb. Phil. Soc. 149 (2010), 455490.CrossRefGoogle Scholar
[5]Adem, A. and Milgram, R. J.Cohomology of Finite Groups (Springer-Verlag Grundlehren 309, 2004).CrossRefGoogle Scholar
[6]Baird, T.Cohomology of the space of commuting n-tuples in a compact Lie group. Algebr. Geom. Topol. 7 (2007), 737754.CrossRefGoogle Scholar
[7]Belyi, V.On Galois extensions of a maximal cyclotomic field. Math. USSR Izv. 14 (1980), 247256.CrossRefGoogle Scholar
[8]Brown, K.Cohomology of Groups (Springer-Verlag GTM 87, 1982).CrossRefGoogle Scholar
[9]Dror-Farjoun, E.Cellular Spaces, Null Spaces and Homotopy Localization (Springer LNM 1622, 1996).Google Scholar
[10]Feit, W. and Thompson, J. G.Solvability of groups of odd order. Pacific J. Math. 13 (1963), 7751029.CrossRefGoogle Scholar
[11]Hatcher, A.Algebraic Topology (Cambridge University Press, 2002).Google Scholar
[12]Ihara, Y. On the embedding of Gal(/ℚ) in. In The Grothendieck Theory of Dessins d'Enfants. London Mathematical Society Lecture Notes Vol. 200 (Cambridge University Press, 1994).Google Scholar
[13]Mac Lane, S. The Milgram bar construction as a tensor product of functors. In Lecture Notes in Math. vol. 168 (Springer, 1970), pp. 135152.Google Scholar
[14]May, J. P.E spaces, group completions, and permutative categories. New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pp. 6193. London Math. Soc. Lecture Note Ser., No. 11 (Cambridge University Press, 1974).CrossRefGoogle Scholar
[15]McCleary, J.A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Math. vol. 58 (Cambridge University Press, 2001).Google Scholar
[16]Milgram, R. J.The bar construction and abelian H-spaces. Ill. J. Math. 11 (1967), 242250.Google Scholar
[17]Milnor, J.The geometrical realization of a semi-simplicial complex. Ann. of Math. 65 (1957), 357362.CrossRefGoogle Scholar
[18]Pakianathan, J. and Yalcin, E.On commuting and noncommuting complexes. J. Algebra 236 (2001), 396418.CrossRefGoogle Scholar
[19]Puppe, V.A remark on homotopy fibrations. Manuscripta Math. 12 (1974), 113120.CrossRefGoogle Scholar
[20]Quillen, D. and Venkov, B. B.Cohomology of finite groups and elementary abelian subgroups. Topology 11 (1972), 317318.CrossRefGoogle Scholar
[21]Schmidt, R.Subgroup lattices of groups. de Gruyter Expositions in Math. 14 (1994).Google Scholar
[22]Serre, J.-P.Trees (Springer-Verlag, 1980).CrossRefGoogle Scholar
[23]Snaith, V.Algebraic cobordism and K-theory. Mem. Amer. Math. Soc. 221 (1979).Google Scholar
[24]Solomon, R.A brief history of the classification of the finite simple groups. Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 315352.CrossRefGoogle Scholar
[25]Steenrod, N. E.Milgram's classifying space of a topological group. Topology 7 (1968), 349368.CrossRefGoogle Scholar
[26]Suzuki, M.The nonexistence of a certain type of simple groups of odd order. Proc. Amer. Math. Soc. 8 (1957), 686695.CrossRefGoogle Scholar
[27]Torres-Giese, E. Spaces of homomorphisms and group cohomology. Ph.D. thesis. University of British Columbia (2007).Google Scholar
[28]Welker, V., Ziegler, G. and Živaljević, R.Homotopy colimits-comparison lemmas for combinatorial applications. J. Reine Angew. Math. 509 (1999), 117149.Google Scholar