String topology

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 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)).

• 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) harvtxt error: no target: CITEREFMeier2010 (help)).

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]

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)).