String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by.

Motivation

While the singular cohomology of a space has always a product structure, this is not true for the singular homology of a space. Nevertheless, it is possible to construct such a structure for an oriented manifold M of dimension d. This is the so-called intersection product. Intuitively, one can describe it as follows: given classes x\in H_p(M) and y\in H_q(M), take their product and make it transversal to the diagonal. The intersection is then a class in, the intersection product of x and y. One way to make this construction rigorous is to use stratifolds. Another case, where the homology of a space has a product, is the (based) loop space \Omega X of a space X. Here the space itself has a product by going first through the first loop and then through the second one. There is no analogous product structure for the free loop space LX of all maps from S^1 to X since the two loops need not have a common point. A substitute for the map m is the map where is the subspace of LM\times LM, where the value of the two loops coincides at 0 and \gamma is defined again by composing the loops.

The Chas–Sullivan product

The idea of the Chas–Sullivan product is to now combine the product structures above. Consider two classes and. Their product x\times y lies in. We need a map One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from LM\times LM to the Thom space of the normal bundle of. Composing the induced map in homology with the Thom isomorphism, we get the map we want. Now we can compose i^! with the induced map of \gamma to get a class in, the Chas–Sullivan product of x and y (see e.g. ).

Remarks

The Batalin–Vilkovisky structure

There is an action by rotation, which induces a map Plugging in the fundamental class, gives an operator of degree 1. One can show that this operator interacts nicely with the Chas–Sullivan product in the sense that they form together the structure of a Batalin–Vilkovisky algebra on. This operator tends to be difficult to compute in general. The defining identities of a Batalin-Vilkovisky algebra were checked in the original paper "by pictures." A less direct, but arguably more conceptual way to do that could be by using an action of a cactus operad on the free loop space LM. The cactus operad is weakly equivalent to the framed little disks operad and its action on a topological space implies a Batalin-Vilkovisky structure on homology.

Field theories

There are several attempts to construct (topological) field theories via string topology. The basic idea is to fix an oriented manifold M and associate to every surface with p incoming and q outgoing boundary components (with n\geq 1) an operation which fulfills the usual axioms for a topological field theory. The Chas–Sullivan product is associated to the pair of pants. It can be shown that these operations are 0 if the genus of the surface is greater than 0.

Sources