Stable splittings, spaces of representations and almost commuting elements in Lie groups

Abstract

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\Rep(\Z^n, SU)$, $\Rep(\Z^n,U)$ and $\Rep(\Z^n, Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.

37 Pages. To appear in Math. Proc. Camb. Phil. Soc

References in corpus (2)

• Cohomology of the space of commuting n-tuples in a compact Lie group
• Fundamental Groups of Commuting Elements in Lie Groups

Cited by in corpus (13)

• A classifying space for commutativity in Lie groups
• On spaces of commuting elements in Lie groups
• Poincare series of character varieties for nilpotent groups
• Commuting elements in reductive groups and Higgs bundles on abelian varieties
• The spectrum for commutative complex $K$-theory
• Cosimplicial Groups and Spaces of Homomorphisms
• Decompositions of suspensions of spaces involving polyhedral products
• Representation spaces for central extensions and almost commuting unitary matrices
• On the second homotopy group of spaces of commuting elements in Lie groups
• Commuting matrices and Atiyah's Real K-theory
• Non-ergodicity on SU(2) and SU(3) character varieties of the once-punctured torus
• On the moduli spaces of commuting elements in the projective unitary groups
• Topological components of spaces of commuting elements in connected nilpotent Lie groups