String topology in three flavors
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String topology in three flavors. / Naef, Florian; Rivera, Manuel; Wahl, Nathalie.
In: EMS Surveys in Mathematical Sciences, Vol. 10, 2023, p. 243-305.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - String topology in three flavors
AU - Naef, Florian
AU - Rivera, Manuel
AU - Wahl, Nathalie
N1 - Publisher Copyright: © 2023 The Author(s).
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Hochschild homology
KW - Loop spaces
KW - string topology
U2 - 10.4171/EMSS/72
DO - 10.4171/EMSS/72
M3 - Journal article
AN - SCOPUS:85170499546
VL - 10
SP - 243
EP - 305
JO - EMS Surveys in Mathematical Sciences
JF - EMS Surveys in Mathematical Sciences
SN - 2308-2151
ER -
ID: 382450375