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 Chas and Sullivan in 1999 (see Chas & Sullivan 1999).

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: \Omega X\times \Omega X \to \Omega X

by going first the first loop and then the second. 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: {\rm Map}(8,M)\to LM

where Map(8, 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. (Here "8" denotes the topological space "figure 8", i.e. the wedge of two circles.)

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^!: H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm Map}(8,M)).

One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting Map(8,M) \subset LM\times LM as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from LM x LM to the Thom space of the normal bundle of Map(8, 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. Cohen & Jones 2002).

Remarks

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: 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.

Field theories

The pair of pants

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 (see Tamanoi2010)

A more structured approach (exhibited in Godin2008) gives \mathcal{}(H_*(LM), H_*(LM)) the structure of a degree d open-closed homological conformal field theory (HCFT) with positive boundary. Ignoring the open-closed part, this amounts to the following structure: let S be a surface with boundary, where the boundary circles are labeled as incoming or outcoming. If there are p incoming and q outgoing and n\geq 1, we get operations

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

parametrized by a certain twisted homology of the mapping class group of S.

References

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