{
  "id": "1710.05366",
  "version": "v1",
  "published": "2017-10-15T17:42:53.000Z",
  "updated": "2017-10-15T17:42:53.000Z",
  "title": "Multiplicative Structure in the Stable Splitting of $Ω SL_n(\\mathbb{C})$",
  "authors": [
    "Jeremy Hahn",
    "Allen Yuan"
  ],
  "comment": "26 pages. Comments welcome!",
  "categories": [
    "math.AT",
    "math.AG",
    "math.RT"
  ],
  "abstract": "The space of based loops in $SL_n(\\mathbb{C})$, also known as the affine Grassmannian of $SL_n(\\mathbb{C})$, admits an $\\mathbb{E}_2$ or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell--Richter splitting is coherently multiplicative, but not $\\mathbb{E}_2$. Nonetheless, we show that the splitting becomes $\\mathbb{E}_2$ after base-change to complex cobordism. Our proof of the $\\mathbb{A}_\\infty$ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-15T17:42:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiplicative structure",
      "stable splitting",
      "infinite wedge sum",
      "loop space stably splits",
      "obstruction-theoretic computations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}