{{Short description|Branch of topology}} {{Technical|date=March 2022}} '''String topology''', a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by {{harvs|txt|first1=Moira|last1=Chas|first2=Dennis|last2=Sullivan|authorlink2=Dennis Sullivan|year=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 <math>M</math> of dimension <math>d</math>. This is the so-called intersection product. Intuitively, one can describe it as follows: given classes <math>x\in H_p(M)</math> and <math>y\in H_q(M)</math>, take their product <math>x\times y \in H_{p+q}(M\times M)</math> and make it transversal to the diagonal <math>M\hookrightarrow M\times M</math>. The intersection is then a class in <math>H_{p+q-d}(M)</math>, the intersection product of <math>x</math> and <math>y</math>. 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 <math>\Omega X</math> of a space <math>X</math>. Here the space itself has a product :<math>m\colon \Omega X\times \Omega X \to \Omega X</math> by going first through the first loop and then through the second one. There is no analogous product structure for the free loop space <math>LX</math> of all maps from <math>S^1</math> to <math>X</math> since the two loops need not have a common point. A substitute for the map <math>m</math> is the map :<math>\gamma\colon {\rm Map}(S^1 \lor S^1, M)\to LM</math> where <math>{\rm Map}(S^1 \lor S^1, M)</math> is the subspace of <math>LM\times LM</math>, where the value of the two loops coincides at 0 and <math>\gamma</math> 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 <math>x\in H_p(LM)</math> and <math>y\in H_q(LM)</math>. Their product <math>x\times y</math> lies in <math>H_{p+q}(LM\times LM)</math>. We need a map :<math>i^!\colon H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm Map}(S^1 \lor S^1,M)).</math> One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting <math>{\rm Map}(S^1 \lor S^1, M) \subset LM\times LM</math> as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from <math>LM\times LM</math> to the Thom space of the normal bundle of <math>{\rm Map}(S^1 \lor S^1, M)</math>. Composing the induced map in homology with the Thom isomorphism, we get the map we want.
Now we can compose <math>i^!</math> with the induced map of <math>\gamma</math> to get a class in <math>H_{p+q-d}(LM)</math>, the Chas–Sullivan product of <math>x</math> and <math>y</math> (see e.g. {{harvtxt|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 <math>H</math> by another multiplicative homology theory <math>h</math> if <math>M</math> is oriented with respect to <math>h</math>. *Furthermore, we can replace <math>LM</math> by <math>L^n M = {\rm Map}(S^n, M)</math>. By an easy variation of the above construction, we get that <math>\mathcal{}h_*({\rm Map}(N,M))</math> is a module over <math>\mathcal{}h_*L^n M</math> if <math>N</math> is a manifold of dimensions <math>n</math>. *The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle <math>{\rm ev}\colon LM\to M</math> with fiber <math>\Omega M</math> and the fiber bundle <math>LE\to LB</math> for a fiber bundle <math>E\to B</math>, which is important for computations (see {{harvtxt|Cohen|Jones|Yan|2004}} and {{harvtxt|Meier|2010}}).
==The Batalin–Vilkovisky structure== There is an action <math>S^1\times LM \to LM</math> by rotation, which induces a map :<math>H_*(S^1)\otimes H_*(LM) \to H_*(LM)</math>. Plugging in the fundamental class <math>[S^1]\in H_1(S^1)</math>, gives an operator :<math>\Delta\colon H_*(LM)\to H_{*+1}(LM)</math> 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 <math>\mathcal{}H_*(LM)</math>. 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 <math>LM</math>.<ref>{{cite conference |url=https://bookstore.ams.org/pspum-73 |title= Notes on universal algebra |last1= Voronov |first1= Alexander |date= 2005 |publisher= Amer. Math. Soc. |book-title= Graphs and Patterns in Mathematics and Theoretical Physics (M. Lyubich and L. Takhtajan, eds.) |pages= 81–103|location= Providence, RI }}</ref> The cactus operad is weakly equivalent to the framed little disks operad<ref>{{cite book |last1= Cohen |first1= Ralph L. |last2= Hess |first2= Kathryn |last3= Voronov |first3= Alexander A. |date= 2006 |title= String topology and cyclic homology |url= http://www.springer.com/birkhauser/mathematics/book/978-3-7643-2182-6 |location= Basel |publisher= Birkhäuser |isbn= 978-3-7643-7388-7 |chapter= The cacti operad}}</ref> and its action on a topological space implies a Batalin-Vilkovisky structure on homology.<ref>{{cite journal |last1= Getzler |first1= Ezra |date= 1994 |title= Batalin-Vilkovisky algebras and two-dimensional topological field theories |url= https://projecteuclid.org/euclid.cmp/1104254599 |journal= Comm. Math. Phys. |volume= 159 |issue= 2 |pages= 265–285 |doi= 10.1007/BF02102639 |arxiv= hep-th/9212043|bibcode= 1994CMaPh.159..265G |s2cid= 14823949 }}</ref>
==Field theories== thumb|right| 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 <math>M</math> and associate to every surface with <math>p</math> incoming and <math>q</math> outgoing boundary components (with <math>n\geq 1</math>) an operation :<math>H_*(LM)^{\otimes p} \to H_*(LM)^{\otimes q}</math> 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 ({{harvtxt|Tamanoi|2010}}).
==References== {{Reflist}}
===Sources=== * {{cite arXiv |last1=Chas |first1=Moira|last2=Sullivan|first2=Dennis|authorlink2= Dennis Sullivan|date=1999 |title=String Topology |eprint=math/9911159v1}} * {{Cite journal|last1=Cohen|first1=Ralph L. | authorlink1=Ralph Louis Cohen|last2= Jones| first2=John D. S. |title=A homotopy theoretic realization of string topology|journal=Mathematische Annalen|volume= 324|pages=773–798|year=2002|issue=4 |mr=1942249|doi=10.1007/s00208-002-0362-0|arxiv=math/0107187|s2cid=16916132 }} * {{cite book |first1=Ralph Louis |last1=Cohen |authorlink=Ralph Louis Cohen |first2=John D. S. |last2=Jones |first3=Jun |last3=Yan |chapter=The loop homology algebra of spheres and projective spaces |title=Categorical decomposition techniques in algebraic topology: International Conference in Algebraic Topology, Isle of Skye, Scotland, June 2001 |editor1-first=Gregory |editor1-last=Arone |editor2-first=John |editor2-last=Hubbuck |editor3-first=Ran |editor3-last=Levi |editor4-first=Michael |editor4-last=Weiss |editor4-link=Michael Weiss (mathematician) |publisher=Birkhäuser |pages=77–92 |year=2004}} * {{Cite journal|last=Meier|first= Lennart| title=Spectral Sequences in String Topology|journal= Algebraic & Geometric Topology|volume= 11 | year=2011 | issue= 5|pages= 2829–2860|doi=10.2140/agt.2011.11.2829|mr=2846913|arxiv=1001.4906|s2cid= 58893087}} * {{Cite journal|first=Hirotaka|last=Tamanoi|title=Loop coproducts in string topology and triviality of higher genus TQFT operations|journal= Journal of Pure and Applied Algebra|volume= 214|issue=5|pages=605–615|year=2010|mr=2577666 |doi=10.1016/j.jpaa.2009.07.011|arxiv=0706.1276|s2cid=2147096}}
Category:Geometric topology Category:Algebraic topology Category:String theory