A fundamental geometric object derived from an arbitrary topological space \(M\) with a marked point \(\star\) is the space of loops in \(M\) based at \(\star\). The Pontryagin algebra \(A\) of \((M,\star)\) is the singular homology of this loop space with the graded algebra structure induced by the standard multiplication of loops. When \(M\) is a smooth oriented manifold with boundary and \(\star\) is chosen on \(\partial M\), the authors define an “intersection” operation \(A\otimes A \to A\otimes A\).

The authors prove that this operation is a double bracket in the sense of Michel Van den Bergh satisfying a version of the Jacobi identity. The authors show that their double bracket induces Gerstenhaber brackets in the representation algebras of \(A\). These results extend the authors' previous work on surfaces, where \(A\) is the group algebra of the fundamental group of a surface and the Gerstenhaber brackets in question are the usual Poisson brackets on the moduli spaces of representations of such a group.

The present work is inspired by the results of William Goldman on surfaces and by the ideas of string topology due to Moira Chas and Dennis Sullivan.