JournalsjncgVol. 13, No. 1pp. 297–361

Singular Hochschild cohomology and algebraic string operations

  • Manuel Rivera

    University of Miami, Coral Gables, USA
  • Zhengfang Wang

    Peking University, Beijing, China
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Given a differential graded (dg) symmetric Frobenius algebra AA we construct an unbounded complex D(A,A)\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 D(A,A)\mathcal{D}^*(A,A) computes the singular Hochschild cohomology of AA. We construct a cyclic (or Calabi–Yau) AA-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an LL-infinity algebra structure extending the classical Gerstenhaber bracket, on D(A,A)\mathcal{D}^*(A,A). Moreover, we prove that the cohomology algebra H(D(A,A))H^*(\mathcal{D}^*(A,A)) is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

Cite this article

Manuel Rivera, Zhengfang Wang, Singular Hochschild cohomology and algebraic string operations. J. Noncommut. Geom. 13 (2019), no. 1, pp. 297–361

DOI 10.4171/JNCG/325