Brackets in Pontryagin algebras of manifolds

Abstract : Given a smooth oriented manifold $M$ with non-empty boundary, we study the Pontryagin algebra $A=H_\ast(\Omega )$ where $ \Omega $ is the space of loops in $M$ based at a distinguished point of $ \partial M$. Using the ideas of string topology of Chas-Sullivan, we define a linear map $\{\{-,-\}\}: A \otimes A \to A\otimes A$ which is a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For $\dim(M)\geq 3$, the double bracket $\{\{-,-\}\}$ induces Gerstenhaber brackets in the representation algebras associated with $A$. This extends our previous work on the case $\dim(M)=2$ where $A= H_0(\Omega )$ is the group algebra of the fundamental group $\pi_1(M)$ and the double bracket $\{\{-,-\}\}$ induces the standard Poisson brackets on the moduli spaces of representations of $\pi_1(M)$.
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https://hal.archives-ouvertes.fr/hal-00853982
Contributor : Gwénaël Massuyeau <>
Submitted on : Monday, August 26, 2013 - 9:38:49 AM
Last modification on : Thursday, August 1, 2019 - 3:18:19 PM

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  • HAL Id : hal-00853982, version 1
  • ARXIV : 1308.5131

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Gwenael Massuyeau, Vladimir Turaev. Brackets in Pontryagin algebras of manifolds. Mémoires de la Société Mathématique de France, Society Math De France, 2017, 154. ⟨hal-00853982⟩

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