Singular hochschild cohomology and algebraic string operations

Manuel Rivera, Zhengfang Wang

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)297-361
Number of pages65
JournalJournal of Noncommutative Geometry
Volume13
Issue number1
DOIs
StatePublished - Jan 1 2019

Keywords

  • A-infinity algebras
  • Frobenius algebras
  • L-infinity algebras
  • String topology
  • Tate–Hochschild complex

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Mathematical Physics
  • Geometry and Topology

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