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).
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 {\displaystyle M} of dimension d {\displaystyle d} . This is the so-called intersection product. Intuitively, one can describe it as follows: given classes x ∈ H p ( M ) {\displaystyle x\in H_{p}(M)} and y ∈ H q ( M ) {\displaystyle y\in H_{q}(M)} , take their product x × y ∈ H p + q ( M × M ) {\displaystyle x\times y\in H_{p+q}(M\times M)} and make it transversal to the diagonal M ↪ M × M {\displaystyle M\hookrightarrow M\times M} . The intersection is then a class in H p + q − d ( M ) {\displaystyle H_{p+q-d}(M)} , the intersection product of x {\displaystyle x} and y {\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 Ω X {\displaystyle \Omega X} of a space X {\displaystyle X} . Here the space itself has a product
m : Ω X × Ω X → Ω X {\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 L X {\displaystyle LX} of all maps from S 1 {\displaystyle S^{1}} to X {\displaystyle X} since the two loops need not have a common point. A substitute for the map m {\displaystyle m} is the map
γ : M a p ( S 1 ∨ S 1 , M ) → L M {\displaystyle \gamma \colon {\rm {Map}}(S^{1}\lor S^{1},M)\to LM}
where M a p ( S 1 ∨ S 1 , M ) {\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)} is the subspace of L M × L M {\displaystyle LM\times LM} , where the value of the two loops coincides at 0 and γ {\displaystyle \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 ∈ H p ( L M ) {\displaystyle x\in H_{p}(LM)} and y ∈ H q ( L M ) {\displaystyle y\in H_{q}(LM)} . Their product x × y {\displaystyle x\times y} lies in H p + q ( L M × L M ) {\displaystyle H_{p+q}(LM\times LM)} . We need a map
i ! : H p + q ( L M × L M ) → H p + q − d ( M a p ( S 1 ∨ S 1 , M ) ) . {\displaystyle i^{!}\colon H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm {Map}}(S^{1}\lor S^{1},M)).}
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