Abstract
This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adem A., Cohen D. and Cohen F.R. (2003). On representations and K-theory of the braid groups. Math Annalen 326: 515–542
Akbulut, S., McCarthy, J.: Casson’s Invariant for Oriented Homology Spheres. Mathematical Notes 36, Princeton University Press (1990)
Borel A., Friedman R. and Morgan J. (2002). Almost commuting elements in compact Lie groups. Mem. Am. Math. Soc. 157(747): x+136
Cohen F.R. (1993). On the mapping class groups for punctured spheres, the hyperelliptic mapping class groups, SO(3) and Spinc(3). Am. J. Math. 115(2): 389–434
Dold, A.: Lectures on Algebraic Topology, Springer, Grundlehren 200 (1980)
Fricke, R., Klein, F.: Vorlesungen über die Theorie der automorphen Funktionen, Bd. 1, Teubner, Leipzig, 1897, 2nd ed. Springer, Grundlehren 200 (1926)
Goldman W.M. (1988). Topological components of the space of representations. Invent Math. 93(3): 557–607
Humphries, J.: Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer, New York (1978)
Huo N. and Liu C.C. (2003). Connected components of surface group representations. Int. Math. Res. Not. 44: 2359–2372
Kac, V.G., Smilga, A.V.:Vacuum structure in supersymmetric Yang-Mills theories with any gauge group, The many faces of the superworld, pp.185–234, World Scientfic. Publishing, River Edge (2000)
Kapovich, M., Millson, J.J.: On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Publications Mathematiques IHES, Vol.88 pp. 5–95 (1998)
Lannes, J.: Théorie homotopique des groupes de Lie (d’aprés W. G. Dwyer et C. W. Wilkerson), Séminaire Bourbaki, Vol. 1993/1994. Astérisque (227 1995), Exp. No. 776, 3, 21–45.
Li J. (1993). The space of surface group representations. Manuscript Mathematics 78(3): 223–243
May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics 171, (1972)
Springer, T.A., Steinberg, R.: Conjugacy Classes, Seminar on Algebraic Groups and Related Finite Groups (IAS Princeton 1968/69), pp. 167–266 in Lecture Notes in Mathematics, Vol. 131 Springer, Berlin (1970)
Steenrod N.E. (1967). A convenient category of topological spaces. Michigan Math J. 14: 133–152
Wajnryb B. (1983). A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45(2-3): 157–174
Author information
Authors and Affiliations
Corresponding author
Additional information
Alejandro Adem was partially supported by the NSF and NSERC. Frederick R. Cohen was partially supported by the NSF, grant number 0340575.
An erratum to this article can be found online at http://dx.doi.org/10.1007/s00208-009-0423-8.
Rights and permissions
About this article
Cite this article
Adem, A., Cohen, F.R. Commuting elements and spaces of homomorphisms. Math. Ann. 338, 587–626 (2007). https://doi.org/10.1007/s00208-007-0089-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0089-z