{
  "id": "2502.05276",
  "version": "v1",
  "published": "2025-02-07T19:19:57.000Z",
  "updated": "2025-02-07T19:19:57.000Z",
  "title": "Homotopy Types of Small Semigroups",
  "authors": [
    "Dennis Sweeney"
  ],
  "comment": "31 pages, 5 Figures",
  "categories": [
    "math.AT"
  ],
  "abstract": "We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most $8$ and present some particular semigroups with notable classifying spaces, refuting conjectures of Nico. Along the way, we give an alternative topological proof of the fact that if a finite semigroup $S$ has a left-simple or right-simple minimal ideal $K(S)$, then the classifying space $BS$ is homotopy equivalent to the classifying space $B(GS)$ of the group completion. We also describe an algorithm for computing the group completion $GS$ of a finite semigroup $S$ using asymptotically fewer than $|S|^2$ semigroup operations. Finally, we show that the set of homotopy types of classifying spaces of finite monoids is closed under suspension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-07T19:19:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M50",
      "55U10",
      "55-08",
      "55-11"
    ],
    "keywords": [
      "homotopy types",
      "small semigroups",
      "classifying space",
      "finite semigroup",
      "group completion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}