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 $$M$$ 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 $$x\times y \in H_{p+q}(M\times M)$$ and make it transversal to the diagonal $$M\hookrightarrow M\times M$$. The intersection is then a class in $$H_{p+q-d}(M)$$, 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
 * $$m\colon \Omega X\times \Omega X \to \Omega X$$

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
 * $$\gamma\colon {\rm Map}(S^1 \lor S^1, M)\to LM$$

where $${\rm Map}(S^1 \lor S^1, M)$$ 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 $$x\in H_p(LM)$$ and $$y\in H_q(LM)$$. Their product $$x\times y$$ lies in $$H_{p+q}(LM\times LM)$$. We need a map
 * $$i^!\colon H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm Map}(S^1 \lor S^1,M)).$$

One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting $${\rm Map}(S^1 \lor S^1, M) \subset LM\times LM$$ 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 $${\rm Map}(S^1 \lor S^1, M)$$. 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 $$H_{p+q-d}(LM)$$, the Chas–Sullivan product of $$x$$ and $$y$$ (see e.g. ).

Remarks

 * As in the case of the intersection product, there are different sign conventions concerning the Chas–Sullivan product. In some convention, it is graded commutative, in some it is not.
 * The same construction works if we replace $$H$$ by another multiplicative homology theory $$h$$ if $$M$$ is oriented with respect to $$h$$.
 * Furthermore, we can replace $$LM$$ by $$L^n M = {\rm Map}(S^n, M)$$. By an easy variation of the above construction, we get that $$\mathcal{}h_*({\rm Map}(N,M))$$ is a module over $$\mathcal{}h_*L^n M$$ if $$N$$ is a manifold of dimensions $$n$$.
 * The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle $${\rm ev}\colon LM\to M$$ with fiber $$\Omega M$$ and the fiber bundle $$LE\to LB$$ for a fiber bundle $$E\to B$$, which is important for computations (see and ).

The Batalin–Vilkovisky structure
There is an action $$S^1\times LM \to LM$$ by rotation, which induces a map
 * $$H_*(S^1)\otimes H_*(LM) \to H_*(LM)$$.

Plugging in the fundamental class $$[S^1]\in H_1(S^1)$$, gives an operator
 * $$\Delta\colon H_*(LM)\to H_{*+1}(LM)$$

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 $$\mathcal{}H_*(LM)$$. 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
 * $$H_*(LM)^{\otimes p} \to H_*(LM)^{\otimes q}$$

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.