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## Summary

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).

## 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. Cohen & Jones (2002)).

## 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 Cohen, Jones & Yan (2004) and Meier (2010)).

## 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

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 (Tamanoi (2010)).