String topology explained

String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by .

Motivation

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 x y \in H _{p+q} (M x M) and make it transversal to the diagonal M\hookrightarrow M x 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.\Omega X of a space X . Here the space itself has a product m\colon \Omega X x \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 x 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 x y lies in H _{p+q} (LM x LM) . We need a mapi ^{!\colon} H _{p+q} (LM x 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 x 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 {\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.).

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 l{}h _{*({\rm} Map}(N,M)) is a module over l{}h ^{n} _{*L} M if N is a manifold of dimensions n .{\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 and).

The Batalin–Vilkovisky structure

There is an action

S ^{1 x} LM \to LM by rotation, which induces a map H ^{1 ) ⊗} _{*(S} H _{*(LM)} \to H _{*(LM)} .Plugging in the fundamental class [S ^{1]\in} H ^{1 )} _{1 (S} , 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 l{}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 .^[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

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 operationH ^{⊗ p} _{*(LM)} \to H ^{⊗ q} _{*(LM)} 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 .

References

Sources

• Chas . Moira. Sullivan. Dennis. Dennis Sullivan. 1999 . String Topology . math/9911159v1.
• Cohen. Ralph L. . Ralph Louis Cohen. Jones. John D. S. . A homotopy theoretic realization of string topology. Mathematische Annalen. 324. 773–798. 2002. 4 . 1942249. 10.1007/s00208-002-0362-0. math/0107187. 16916132 .
• Book: Ralph Louis Cohen

. Ralph Louis . Cohen . Ralph Louis Cohen . John D. S. . Jones . Jun . Yan . The loop homology algebra of spheres and projective spaces . Categorical decomposition techniques in algebraic topology: International Conference in Algebraic Topology, Isle of Skye, Scotland, June 2001 . Gregory . Arone . John . Hubbuck . Ran . Levi . Michael . Weiss . Michael Weiss (mathematician) . . 77–92 . 2004.

• Meier. Lennart. Spectral Sequences in String Topology. Algebraic & Geometric Topology. 11 . 2011 . 5. 2829–2860. 10.2140/agt.2011.11.2829. 2846913. 1001.4906. 58893087.
• Hirotaka. Tamanoi. Loop coproducts in string topology and triviality of higher genus TQFT operations. Journal of Pure and Applied Algebra. 214. 5. 605–615. 2010. 2577666 . 10.1016/j.jpaa.2009.07.011. 0706.1276. 2147096.

Notes and References

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