String topology

String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces.

The field was started by Moira Chas and Dennis Sullivan (1999).

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

This is the so-called intersection product.

Intuitively, one can describe it as follows: given classes

and make it transversal to the diagonal

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

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

since the two loops need not have a common point.

is defined again by composing the loops.

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

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.

by rotation, which induces a map Plugging in the fundamental class

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

[1] The cactus operad is weakly equivalent to the framed little disks operad[2] and its action on a topological space implies a Batalin-Vilkovisky structure on homology.

[3] There are several attempts to construct (topological) field theories via string topology.

The basic idea is to fix an oriented manifold

outgoing boundary components (with

) 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 (Tamanoi (2010)).