{
  "id": "1703.03899",
  "version": "v1",
  "published": "2017-03-11T03:25:16.000Z",
  "updated": "2017-03-11T03:25:16.000Z",
  "title": "Singular Hochschild cohomology and algebraic string operations",
  "authors": [
    "Manuel Rivera",
    "Zhengfang Wang"
  ],
  "comment": "48 pages, 9 figures, preliminary version, comments are welcome",
  "categories": [
    "math.RT",
    "math.AT",
    "math.KT",
    "math.QA"
  ],
  "abstract": "Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gerstenhaber bracket, on $\\mathcal{D}^*(A,A)$. Moreover, we prove that the cohomology algebra $H^*(\\mathcal{D}^*(A,A))$ is a Batalin-Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two Frobenius algebras are quasi-isomorphic as dg algebras then their Tate-Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate-Hochschild complex to string topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-11T03:25:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular hochschild cohomology",
      "algebraic string operations",
      "infinity algebra structure",
      "tate-hochschild complex",
      "symmetric frobenius algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}