Brackets in the 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 $\dim (M)\ge 3$, 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 ${\pi }_1(M)$.

114 pages. Reorganization of the paper, and some minor modifications in the text