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Given a differential graded (dg) symmetric Frobenius algebra we construct an unbounded complex , 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 computes the singular Hochschild cohomology of . We construct a cyclic (or Calabi–Yau) -infinity algebra structure, which extends the classical Hochschild cup and cap products, and an -infinity algebra structure extending the classical Gerstenhaber bracket, on . Moreover, we prove that the cohomology algebra 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.
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Manuel Rivera, Zhengfang Wang, Singular Hochschild cohomology and algebraic string operations. J. Noncommut. Geom. 13 (2019), no. 1, pp. 297–361