# Brackets in the Pontryagin Algebras of Manifolds

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*Gwénaël Massuyeau; Vladimir Turaev*

A publication of the Société Mathématique de France

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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians.