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).

MotivationEdit

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   of dimension  . This is the so-called intersection product. Intuitively, one can describe it as follows: given classes   and  , take their product   and make it transversal to the diagonal  . The intersection is then a class in  , the intersection product of   and  . 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   of a space  . Here the space itself has a product

 

by going first through the first loop and then through the second one. There is no analogous product structure for the free loop space   of all maps from   to   since the two loops need not have a common point. A substitute for the map   is the map

 

where   is the subspace of  , where the value of the two loops coincides at 0 and   is defined again by composing the loops.

The Chas–Sullivan productEdit

The idea of the Chas–Sullivan product is to now combine the product structures above. Consider two classes   and  . Their product   lies in  . 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   as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from   to the Thom space of the normal bundle of  . Composing the induced map in homology with the Thom isomorphism, we get the map we want.

Now we can compose   with the induced map of   to get a class in  , the Chas–Sullivan product of   and   (see e.g. Cohen & Jones (2002)).

RemarksEdit

  • 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   by another multiplicative homology theory   if   is oriented with respect to  .
  • Furthermore, we can replace   by  . By an easy variation of the above construction, we get that   is a module over   if   is a manifold of dimensions  .
  • The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle   with fiber   and the fiber bundle   for a fiber bundle  , which is important for computations (see Cohen, Jones & Yan (2004) and Meier (2010)).

The Batalin–Vilkovisky structureEdit

There is an action   by rotation, which induces a map

 .

Plugging in the fundamental class  , 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  . 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  .[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 theoriesEdit

 
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   and associate to every surface with   incoming and   outgoing boundary components (with  ) 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 (Tamanoi (2010)).

ReferencesEdit

  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.

SourcesEdit