Abstract. In this article we consider a space B com G assembled from commuting elements in a Lie group G first defined in [3]. 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 × B com U 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 × B com U ]. We compute the rational cohomology of B com G for G equal to any of the classical groups SU (r), U (q) and Sp(k), and exhibit the rational cohomologies of B com U , B com SU and B com Sp as explicit polynomial rings.

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. ⎞ ⎠ .When Rep(Z n , G) is path-connected for all n 1 then the following theorem completely describes this decomposition:connected Lie group such that Rep(Z n , G) is connected for every n 1. Let T be a maximal torus of G and W the Weyl group associated to Stable splittings and almost commuting elements 457 T . Then Rep(Z n , G) % T n /W with W acting diagonally on T n . Moreover there is a homotopy equivalencewhere T ∧r is the r -fold smash product of T and W acts diagonally on T ∧r .The previous theorem applies in particular to the cases G = U (m), Sp(m) and SU (m). It turns out that for such G the representation spaces Rep(Z n , G) can be identified in terms of symmetric products. PROPOSITION 1·4. There are homeomorphismswhere Z/2 acts diagonally by complex conjugation on each factor. Moreover for each m, n 1 the determinant map defines a locally trivial bundleThis identification provides a number of interesting consequences, for example COROLLARY 1·5. For every m 1 there are homeomorphismsAnother consequence of Proposition 1·4 is the determination of the homotopy type of the inductive limits of the spaces of commuting elements modulo conjugation for the (special) unitary groups and the symplectic groups. Note that this result for the unitary groups is also proved in the recent preprint [22].

Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU , BU , BSO, BO, BSp, BGL ∞ (R) + and Q 0 (S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital E ∞ -ring spectra. We introduce the notion of q-nilpotent K-theory of a CW-complex X for any q ≥ 2, which extends the notion of commutative K-theory defined by Adem-Gómez, and show that it is represented by Z × B(q, U ), were B(q, U ) is the q-th term of the aforementioned filtration of BU .For the proof we introduce an alternative way of associating an infinite loop space to a commutative I-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative I-rig and show that they give rise to non-unital E ∞ -ring spectra.

Abstract. Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected subgroups of maximal rank. Let T ⊂ G be a maximal torus with Weyl group W . If the fixed-point set X T has the homotopy type of a finite W -CW complex, we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W -action on the fixed-point set XT we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.

In this paper the space of commuting elements in the central product G m,p of m copies of the special unitary group SU (p) is studied, where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(Z n , G m,p ) of isomorphism classes of flat connections on principal G m,p -bundles over the n-torus is completely described for all values of n, m and p.