# Singular Hochschild cohomology and algebraic string operations

### Manuel Rivera

University of Miami, Coral Gables, USA### Zhengfang Wang

Peking University, Beijing, China

## Abstract

Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $D_{∗}(A,A)$, 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 $D_{∗}(A,A)$ 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 $D_{∗}(A,A)$. Moreover, we prove that the cohomology algebra $H_{∗}(D_{∗}(A,A))$ 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