Rouché–Capelli theorem

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 in some way 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

• 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 $ev:LM\to M$ 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&Yan2004 and Meier2010).

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

• M. Chas & D. Sullivan, String Topology, arXiv:math/9911159v1 (1999)
• R. Cohen & J. Jones, A homotopy theoretic realization of string topology, Mathematische Annalen 324, p. 773–798 (2002)
• R. Cohen & J. Jones & J. Yan, The loop homology algebra of spheres and projective spaces in Categorical decomposition techniques in algebraic topology: International Conference in Algebraic Topology, Isle of Skye, Scotland, June 2001, Birkhäuser, p. 77–92 (2004).
• L. Meier, Spectral Sequences in String Topology, arXiv:1001.4906v2 (2010)
• V. Godin, Higher string topology operations, arXiv:0711.4859v2 (2008)
• H. Tamanoi, Loop coproducts in string topology and triviality of higher genus TQFT operations , Journal of Pure and Applied Algebra 214, Issue 5 pp. 605–615 (2010)