{
  "id": "1402.6309",
  "version": "v2",
  "published": "2014-02-25T20:41:37.000Z",
  "updated": "2016-06-02T16:28:49.000Z",
  "title": "On spaces of commuting elements in Lie groups",
  "authors": [
    "Frederick R. Cohen",
    "Mentor Stafa"
  ],
  "comment": "27 pages, with an appendix by Vic Reiner",
  "doi": "10.1017/S0305004116000311",
  "categories": [
    "math.AT"
  ],
  "abstract": "The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\\mathbb Z^n,G)$, with the order of the Weyl group inverted.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-25T20:41:37.000Z",
      "abstract": "The purpose of this paper is to introduce a new method of \"stabilizing\" spaces of homomorphisms $Hom(\\pi,G)$ where $\\pi$ 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 $Hom(\\mathbb{Z}^n,G)$, by assembling these spaces into a single space for all $n \\geq 0$. 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 $Hom(\\mathbb{Z}^n,G)$ 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 $Hom(\\mathbb{Z}^n,G)$ 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. The appendix gives the Hilbert-Poincare series of $Comm(G)$.",
      "comment": "28 pages, with an appendix by Vic Reiner",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-06-02T16:28:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E99",
      "20G05"
    ],
    "keywords": [
      "commuting elements",
      "single space",
      "compact lie group",
      "singular homology",
      "maximal torus"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1282955,
      "adsabs": "2014arXiv1402.6309C"
    }
  }
}