Loop spaces and bialgebras

Manuel Rivera, Purdue University

Online Talk 

The perfect pairing of Poincaré duality on a closed oriented manifold of dimension d yields a (graded) commutative and associative degree -d intersection product on the homology of the manifold. Assuming field coefficients, the diagonal approximation coproduct on homology is compatible with the intersection product. This compatibility says that the coproduct is a map of bimodules with respect to the intersection product. This type of bialgebra structure is known as a Frobenius (bi)algebra. The free loop space of a manifold is not a finite dimensional manifold anymore, nonetheless one may combine the intersection product on the underlying manifold together with loop concatenation to construct an intersection type product on homology. This construction is known as the Chas-Sullivan product in string topology. We will discuss different types of bialgebra structures extending the Chas-Sullivan product that may be interpreted as manifestations of classical Poincaré duality on the free loop space of a manifold.