Description

A fundamental geometric object derived from an arbitrary topological space M with a marked point (...) is the space of loops in M based at (...). The Pontryagin algebra A of (M, (...)) 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 (...) is chosen on (...)M, we define an « intersection » operation A (...) A (...) A (...) A. We prove that this operation is a double bracket in the sense of Michel Van den Bergh satisfying a version of the Jacobi identity. We show that our double bracket induces Gerstenhaber brackets in the representation algebras of A. These results extend our 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.