On Spaces of Commuting Elements in Lie Groups

The purpose of this paper is to introduce a new method of “stabilizing” spaces of homomorphisms Hom(π,G) where π is a certain choice of finitely generated group and G is a compact Lie group. The main results apply to the space of all ordered n-tuples of pairwise commuting elements in a compact Lie group G, denoted Hom(Zn, G), by assembling these spaces into a single space for all n ≥ 0. The resulting space denoted Comm(G) is an infinite dimensional analogue of a Stiefel manifold which can be regarded as the space, suitably topologized, of all finite ordered sets of generators for all finitely generated abelian subgroups of G. The methods are to develop the geometry and topology of the free associative monoid generated by a maximal torus of G, and to “twist” this free monoid into a space which approximates the space of “all commuting n-tuples” for all n, Comm(G), into a single space. Thus a new space Comm(G) is introduced which assembles the spaces Hom(Zn, G) into a single space for all positive integers n. Topological properties of Comm(G) are developed while the singular homology of this space is computed with coefficients in the ring of integers with the order of the Weyl group of G inverted. One application is that the cohomology of Hom(Zn, G) follows from that of Comm(G) for any cohomology theory. The results for singular homology of Comm(G) are given in terms of the tensor algebra generated by the reduced homology of a maximal torus. Applications to classical Lie groups as well as exceptional Lie groups are given. A stable decomposition of Comm(G) is also given here with a significantly finer stable decomposition to be given in the sequel to this paper along with extensions of these constructions to additional representation varieties. An appendix by V. Reiner is included which uses the results here concerning Comm(G) together with Molien’s theorem to give the Hilbert-Poincaré series of Comm(G).