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| string topology | |
|---|---|
| Name | String topology |
| Field | Algebraic topology |
| Introduced | 1990s |
| Founders | Chas–Sullivan |
| Key concepts | Loop space, intersection product, BV algebra, Hochschild homology, Floer homology |
| Notable works | Chas–Sullivan 1999, Cohen–Jones 2002, Sullivan lectures |
string topology
String topology studies algebraic and geometric structures on spaces of closed loops in manifolds, examining operations, products, and dualities that arise from intersecting and composing loops. It blends techniques from algebraic topology, differential topology, and homological algebra to reveal rich algebraic structures—such as BV algebras and Gerstenhaber algebras—on the homology of loop spaces. The subject interacts with work of many figures and institutions across geometry and mathematical physics.
String topology originated in investigations of loop space homology and the algebraic operations induced by geometric intersections of loops in manifolds. Key players associated with foundational developments include Moira Chas, Dennis Sullivan, John Milnor, Edward Witten, and Graeme Segal, while important venues for dissemination include Institute for Advanced Study, Mathematical Sciences Research Institute, and Clay Mathematics Institute. Foundational methods draw upon constructions from Morse theory on loop spaces by Marston Morse, techniques from Hassler Whitney-style transversality, and algebraic frameworks such as Gerstenhaber algebra and Batalin–Vilkovisky algebra studied in works by Gerstenhaber and Batalin–Vilkovisky contributors.
Early impetus for string topology came from analogies between closed string interactions in Edward Witten's lectures and algebraic operations on free loop spaces explored by Moira Chas and Dennis Sullivan in the late 1990s. Precedents trace to intersection theory by René Thom, loop product ideas influenced by Raoul Bott and Shoshichi Kobayashi, and algebraic structures in Hochschild theory developed by Gerald Hochschild and applied by Jean-Louis Loday. Seminal publications and talks at institutions like Institute for Advanced Study, Simons Center for Geometry and Physics, and conferences organized by American Mathematical Society and European Mathematical Society helped formalize operations and encourage collaborations involving Ralph Cohen, John Jones, Alexander A. Voronov, Kai Behrend, and Alberto Cattaneo.
String topology reveals algebraic operations on loop homology analogous to algebraic structures studied by Gerstenhaber and Getzler: a loop product analogous to intersection products in René Thom theory, a coproduct reflecting cutting operations akin to constructions in Hochschild cohomology by Jean-Louis Loday, and a BV operator connected to ideas in Batalin–Vilkovisky formalism. Important contributors to formal algebraic frameworks include Ralph Cohen, John Jones, Alexander A. Voronov, Tristan Rivière, and Dennis Sullivan. Categorical and operadic perspectives link to the Deligne conjecture addressed by Pierre Deligne, and to operad theory developed by Murray Gerstenhaber-related authors and M. Kontsevich. Connections to cyclic homology and Connes's work arise through Hochschild–Connes comparisons pursued by Jean L. Loday and Alain Connes.
Geometric foundations use transversality arguments influenced by René Thom and geometric measure techniques related to Lars Hörmander's analytic frameworks. Homological algebra techniques draw on Pierre Deligne-inspired deformation theory, Max Karoubi's cyclic homology, and chain-level models developed by Ralph Cohen, John Jones, and Dennis Sullivan. Floer-theoretic analogues introduced by Andreas Floer and analytical tools from Simon Donaldson and Edward Witten provide compactness and gluing methods adapted to loop space settings; analytic foundations were expanded by teams associated with Instituto Superior Técnico, ETH Zurich, and IHES. Operadic and moduli-space methods, drawing on M. Kontsevich and Maxim Kontsevich's graph complexes and on work by Getzler and Kapranov, furnish chain-level operations and compatibility relations.
Concrete computations in string topology include calculations for spheres, tori, and complex projective spaces derived by researchers such as Ralph Cohen, Johnson, Felix], Thomas Tradler], and Alberto Abbondandolo. Loop homology of spheres and projective spaces exhibits explicit BV and Gerstenhaber structures computed using Morse–Bott techniques from Marston Morse and spectral sequence methods developed by Jean Leray and Henri Cartan. Computational programs at institutions like Max Planck Institute for Mathematics and Centre de Recerca Matemàtica produced examples showing nontrivial string operations on high-dimensional manifolds, with contributors including Kathryn Hess, Dan Petersen, and Ulrich Tillmann.
String topology has deep ties to topological field theories and symplectic topology. Analogies with closed-string field theory discussed by Edward Witten and algebraic structures in Atiyah-style topological quantum field theory promoted by Michael Atiyah underpin categorical interpretations. Symplectic counterparts involve relations to Floer homology by Andreas Floer and wrapped Fukaya categories developed by Paul Seidel and Mohammed Abouzaid, with exact correspondences explored by Kai Cieliebak and Yasha Eliashberg. Dualities and mirror symmetry considerations echo work by Maxim Kontsevich and Stanisław Ulam-adjacent communities, while mathematical physics interactions engage researchers at Perimeter Institute and CERN.
Applications of string topology influence questions in manifold topology, homotopy theory, and mathematical physics, informing classification problems tackled by William Thurston, Michael Freedman, and Grigori Perelman. Open problems include chain-level formality questions related to Deligne conjecture-type statements, comparisons between string topology operations and symplectic field theory of Eliashberg and Givental, and computational challenges in higher-genus operations studied by Dennis Sullivan and Graeme Segal. Active research by groups at University of Chicago, Princeton University, Stanford University, and University of Cambridge continues to address extension to orbifolds, equivariant phenomena linked to Atiyah–Bott localization, and interactions with categorical structures in the work of Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi.