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| Chas–Sullivan | |
|---|---|
| Name | Chas–Sullivan |
| Subject | Mathematical construction in topology |
| Introduced | 1990s |
| Founders | Moira Chas; Dennis Sullivan |
| Field | Algebraic topology; String topology; Homological algebra |
Chas–Sullivan
Chas–Sullivan is a construction in algebraic topology introduced by Moira Chas and Dennis Sullivan that defines a graded associative product on the homology of the free loop space of a manifold, together with related operations yielding a rich algebraic structure. It connects the topology of mapping spaces such as the free loop space to operations familiar from Gerstenhaber algebra, Batalin–Vilkovisky algebra, and Hochschild homology contexts, and it has influenced developments linking Morse theory, Floer homology, Gromov–Witten invariants, and symplectic topology. The construction generated a broad literature intersecting work by researchers at institutions such as Institute for Advanced Study, Princeton University, Stanford University, and University of Chicago.
The origins of the construction trace to collaborative work by Moira Chas and Dennis Sullivan in the late 1990s, motivated by analogies between loop spaces considered by Vladimir Arnold and algebraic structures appearing in string theory studies by physicists at CERN and institutes influenced by Edward Witten. Early formulations built on classical results about the free loop space from Jean-Pierre Serre, Marston Morse, and later formalizations by Ralph Cohen and John Jones, and were informed by operadic ideas from Getzler and Jones (homology) perspectives. Subsequent developments connected to work of Maxim Kontsevich on deformation quantization, Hinich on homotopical algebra, and Costello on topological conformal field theory.
The central Chas–Sullivan product is defined on H_*(ΛM), the homology of the free loop space ΛM of a closed, oriented manifold M, and yields a graded associative product of degree −dim(M). Constructions employ transversality techniques inspired by Morse theory on loop spaces and intersection-theoretic ideas reminiscent of Thom and Pontryagin. Algebraic formulations relate the product to the loop composition map and to pair-of-pants cobordisms studied in the context of Segal-type field theories and work of Atiyah on topological field theory. Variants use chains modeled on cubical or singular chains developed in the lineage of Eilenberg–MacLane and Cartan.
Chas–Sullivan sits at the core of the subject called string topology, which studies algebraic operations on loop spaces and mapping spaces of manifolds. String topology connects to Gromov–Witten invariants by providing classical counterparts to quantum operations appearing in symplectic topology and enumerative geometry, and interfaces with Floer homology via isomorphisms established in work by Abouzaid, Seidel, and Viterbo. Applications span computations in low-dimensional topology involving knot theory and relations to invariants considered by Witten and Reshetikhin–Turaev. The field influenced categorical formulations linking to Fukaya category constructions and to approaches pursued at MIT and Harvard University.
The Chas–Sullivan operations endow H_*(ΛM) with structures such as a graded commutative product, a bracket yielding a Gerstenhaber algebra, and, when paired with a degree-one operator, a Batalin–Vilkovisky algebra structure discovered in work influenced by Getzler and Tamarkin. Algebraic connections link the loop homology with Hochschild homology HH_*(C^*(M),C^*(M)) of the cochain algebra of M, following approaches by Jones and Tradler. Homological properties include compatibility with the cap product and Poincaré duality as formalized by Poincaré and extended in algebraic topology by Spanier–Whitehead duality frameworks; cyclic homology perspectives from Connes also play a role in understanding periodicity phenomena.
Explicit computations for spheres, tori, and complex projective spaces illustrate the theory. For the n-sphere S^n the loop homology algebra reflects classical results of Bott on periodicity and of Serre on homotopy groups; for the torus T^n the product corresponds to the group algebra structure related to Pontryagin duality and lattice computations familiar from work on Kronecker-type flows. Calculations for projective spaces relate to the cohomology descriptions obtained by Hopf and Leray–Hirsch theorems and to quantum cohomology results by Dubrovnik-style authors. Computational techniques exploit spectral sequences such as the Serre spectral sequence and algebraic tools developed by Eilenberg–Moore and Adams.
The formalism aligns closely with two-dimensional field theories: pair-of-pants and moduli of Riemann surfaces studied by Segal and Kontsevich–Soibelman provide geometric sources for the operations, leading to formulations in terms of operads and PROPs introduced by May and Boardman–Vogt. Operadic encodings via the little disks operad and the framed little disks operad relate to BV structures as explored by Getzler and Voronov; connections to topological conformal field theory echo constructions by Costello and Segal.
Extensions include equivariant versions involving S^1-actions building on work by Atiyah–Bott and equivariant cohomology approaches of Borel, as well as categorical lifts to string topology for classifying spaces treated by Félix, Halperin, and Thomas. Generalizations to orbifolds integrate ideas from Chen–Ruan orbifold cohomology, and adaptations to noncompact or open manifolds connect with relative theories developed alongside Floer and Symplectic Field Theory authors. Higher-dimensional and factorization homology perspectives tie Chas–Sullivan-type structures to recent work by Lurie and Francis in the context of higher categories and topological quantum field theories.