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String topology

## 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 ${\displaystyle M}$  of dimension ${\displaystyle d}$ . This is the so-called intersection product. Intuitively, one can describe it as follows: given classes ${\displaystyle x\in H_{p}(M)}$  and ${\displaystyle y\in H_{q}(M)}$ , take their product ${\displaystyle x\times y\in H_{p+q}(M\times M)}$  and make it transversal to the diagonal ${\displaystyle M\hookrightarrow M\times M}$ . The intersection is then a class in ${\displaystyle H_{p+q-d}(M)}$ , the intersection product of ${\displaystyle x}$  and ${\displaystyle y}$ . 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 ${\displaystyle \Omega X}$  of a space ${\displaystyle X}$ . Here the space itself has a product

${\displaystyle m\colon \Omega X\times \Omega X\to \Omega X}$

by going first through the first loop and then through the second one. There is no analogous product structure for the free loop space ${\displaystyle LX}$  of all maps from ${\displaystyle S^{1}}$  to ${\displaystyle X}$  since the two loops need not have a common point. A substitute for the map ${\displaystyle m}$  is the map

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

where ${\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)}$  is the subspace of ${\displaystyle LM\times LM}$ , where the value of the two loops coincides at 0 and ${\displaystyle \gamma }$  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 ${\displaystyle x\in H_{p}(LM)}$  and ${\displaystyle y\in H_{q}(LM)}$ . Their product ${\displaystyle x\times y}$  lies in ${\displaystyle H_{p+q}(LM\times LM)}$ . We need a map

${\displaystyle i^{!}\colon H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm {Map}}(S^{1}\lor S^{1},M)).}$

One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting ${\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)\subset LM\times LM}$  as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from ${\displaystyle LM\times LM}$  to the Thom space of the normal bundle of ${\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)}$ . Composing the induced map in homology with the Thom isomorphism, we get the map we want.

Now we can compose ${\displaystyle i^{!}}$  with the induced map of ${\displaystyle \gamma }$  to get a class in ${\displaystyle H_{p+q-d}(LM)}$ , the Chas–Sullivan product of ${\displaystyle x}$  and ${\displaystyle y}$  (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 ${\displaystyle H}$  by another multiplicative homology theory ${\displaystyle h}$  if ${\displaystyle M}$  is oriented with respect to ${\displaystyle h}$ .
• Furthermore, we can replace ${\displaystyle LM}$  by ${\displaystyle L^{n}M={\rm {Map}}(S^{n},M)}$ . By an easy variation of the above construction, we get that ${\displaystyle {\mathcal {}}h_{*}({\rm {Map}}(N,M))}$  is a module over ${\displaystyle {\mathcal {}}h_{*}L^{n}M}$  if ${\displaystyle N}$  is a manifold of dimensions ${\displaystyle n}$ .
• The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle ${\displaystyle {\rm {ev}}\colon LM\to M}$  with fiber ${\displaystyle \Omega M}$  and the fiber bundle ${\displaystyle LE\to LB}$  for a fiber bundle ${\displaystyle E\to B}$ , which is important for computations (see Cohen, Jones & Yan (2004) and Meier (2010)).

## The Batalin–Vilkovisky structure

There is an action ${\displaystyle S^{1}\times LM\to LM}$  by rotation, which induces a map

${\displaystyle H_{*}(S^{1})\otimes H_{*}(LM)\to H_{*}(LM)}$ .

Plugging in the fundamental class ${\displaystyle [S^{1}]\in H_{1}(S^{1})}$ , gives an operator

${\displaystyle \Delta \colon H_{*}(LM)\to H_{*+1}(LM)}$

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 ${\displaystyle {\mathcal {}}H_{*}(LM)}$ . 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 ${\displaystyle LM}$ .[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]

## 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 ${\displaystyle M}$  and associate to every surface with ${\displaystyle p}$  incoming and ${\displaystyle q}$  outgoing boundary components (with ${\displaystyle n\geq 1}$ ) an operation

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

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)).

## References

1. ^ Voronov, Alexander (2005). "Notes on universal algebra". Graphs and Patterns in Mathematics and Theoretical Physics (M. Lyubich and L. Takhtajan, eds.). Providence, RI: Amer. Math. Soc. pp. 81–103.
2. ^ Cohen, Ralph L.; Hess, Kathryn; Voronov, Alexander A. (2006). "The cacti operad". String topology and cyclic homology. Basel: Birkhäuser. ISBN 978-3-7643-7388-7.
3. ^ Getzler, Ezra (1994). "Batalin-Vilkovisky algebras and two-dimensional topological field theories". Comm. Math. Phys. 159 (2): 265–285. arXiv:hep-th/9212043. doi:10.1007/BF02102639. S2CID 14823949.