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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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1435-5345
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Volume 2016, Issue 720

# Universal operations in Hochschild homology

Nathalie Wahl
Published Online: 2014-07-09 | DOI: https://doi.org/10.1515/crelle-2014-0037

## Abstract

We provide a general method for finding all natural operations on the Hochschild complex of $\mathcal{ℰ}$-algebras, where $\mathcal{ℰ}$ is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative or ${A}_{\mathrm{\infty }}$-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When $\mathcal{ℰ}$ encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich–Soibelman via different methods identify with all formal operations. When $\mathcal{ℰ}$ encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler–Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.

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Revised: 2014-03-11

Published Online: 2014-07-09

Published in Print: 2016-11-01

The author was supported by the Danish National Sciences Research Council (DNSRC) and the European Research Council (ERC), as well as by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 720, Pages 81–127, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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