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 .
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 formula_1 of dimension formula_2. This is the so-called intersection product. Intuitively, one can describe it as follows: given classes formula_3 and formula_4, take their product formula_5 and make it transversal to the diagonal formula_6. The intersection is then a class in formula_7, the intersection product of formula_8 and formula_9. 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 formula_10 of a space formula_11. Here the space itself has a product 
by going first the first loop and then the second. There is no analogous product structure for the free loop space formula_13 of all maps from formula_14 to formula_11 since the two loops need not have a common point. A substitute for the map formula_16 is the map 
where formula_18 is the subspace of formula_19, where the value of the two loops coincides at 0 and formula_20 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 formula_21 and formula_22. Their product formula_23 lies in formula_24. 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 formula_26 as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from formula_19 to the Thom space of the normal bundle of formula_18. Composing the induced map in homology with the Thom isomorphism, we get the map we want.
Now we can compose formula_29 with the induced map of formula_20 to get a class in formula_31, the Chas–Sullivan product of formula_8 and formula_9 (see e.g. ).
The Batalin–Vilkovisky structure.
There is an action formula_48 by rotation, which induces a map 
Plugging in the fundamental class formula_50, 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 formula_52. 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 formula_38. 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 formula_1 and associate to every surface with formula_55 incoming and formula_56 outgoing boundary components (with formula_57) 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 (see )
A more structured approach (exhibited in ) gives formula_59 the structure of a degree formula_2 open-closed homological conformal field theory (HCFT) with positive boundary. Ignoring the open-closed part, this amounts to the following structure: let formula_61 be a surface with boundary, where the boundary circles are labeled as incoming or outcoming. If there are formula_55 incoming and formula_56 outgoing and formula_57, we get operations
parametrized by a certain twisted homology of the mapping class group of formula_61.