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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Lifts of $C_\infty$- and $L_\infty$-morphisms to $G_\infty$-morphisms
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by Grégory Ginot and Gilles Halbout PDF
Proc. Amer. Math. Soc. 134 (2006), 621-630 Request permission


Let $\mathfrak {g}_2$ be the Hochschild complex of cochains on $C^\infty (\mathbb {R}^n)$ and let $\mathfrak {g}_1$ be the space of multivector fields on $\mathbb {R}^n$. In this paper we prove that given any $G_\infty$-structure (i.e. Gerstenhaber algebra up to homotopy structure) on $\mathfrak {g}_2$, and any $C_\infty$-morphism $\varphi$ (i.e. morphism of a commutative, associative algebra up to homotopy) between $\mathfrak {g}_1$ and $\mathfrak {g}_2$, there exists a $G_\infty$-morphism $\Phi$ between $\mathfrak {g}_1$ and $\mathfrak {g}_2$ that restricts to $\varphi$. We also show that any $L_\infty$-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a $G_\infty$-morphism, using Tamarkin’s method for any $G_\infty$-structure on $\mathfrak {g}_2$. We also show that any two of such $G_\infty$-morphisms are homotopic.
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Additional Information
  • Grégory Ginot
  • Affiliation: Laboratoire Analyse Géométrie et Applications, Université Paris 13 et Ecole Normale Supèrieure de Cachan, France
  • Email:
  • Gilles Halbout
  • Affiliation: Institute de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France
  • Email:
  • Received by editor(s): April 1, 2003
  • Received by editor(s) in revised form: September 30, 2004
  • Published electronically: July 18, 2005
  • Communicated by: Paul Goerss
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 621-630
  • MSC (2000): Primary 16E40, 53D55; Secondary 18D50, 16S80
  • DOI:
  • MathSciNet review: 2180877