String topology in three flavors
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We describe two major string topology operations, the Chas-Sullivan product and the Goresky-Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom-Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3-dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate-Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply-connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically.
Originalsprog | Engelsk |
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Tidsskrift | EMS Surveys in Mathematical Sciences |
Vol/bind | 10 |
Sider (fra-til) | 243-305 |
Antal sider | 63 |
ISSN | 2308-2151 |
DOI | |
Status | Udgivet - 2023 |
Bibliografisk note
Funding Information:
Funding. F. N. and N. W. have received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska–Curie (grant agreement no. 896370) and the European Research Council (grant agreement no. 772960), respectively, and were both supported by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151). M. R. was supported by NSF Grant 210554 and the Karen EDGE Fellowship.
Publisher Copyright:
© 2023 The Author(s).
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