String topology in three flavors

Department of Mathematical Sciences

Research output: Contribution to journal › Journal article › Research › peer-review

Standard

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

Harvard

Naef, F, Rivera, M & Wahl, N 2023, 'String topology in three flavors', EMS Surveys in Mathematical Sciences, vol. 10, pp. 243-305. https://doi.org/10.4171/EMSS/72

APA

Naef, F., Rivera, M., & Wahl, N. (2023). String topology in three flavors. EMS Surveys in Mathematical Sciences, 10, 243-305. https://doi.org/10.4171/EMSS/72

Vancouver

Naef F, Rivera M, Wahl N. String topology in three flavors. EMS Surveys in Mathematical Sciences. 2023;10:243-305. https://doi.org/10.4171/EMSS/72

Author

Naef, Florian ; Rivera, Manuel ; Wahl, Nathalie. / String topology in three flavors. In: EMS Surveys in Mathematical Sciences. 2023 ; Vol. 10. pp. 243-305.

Bibtex

@article{62f025b37dc14919938fd44d50d1b881,

title = "String topology in three flavors",

abstract = "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.",

keywords = "Hochschild homology, Loop spaces, string topology",

author = "Florian Naef and Manuel Rivera and Nathalie Wahl",

note = "Publisher Copyright: {\textcopyright} 2023 The Author(s).",

year = "2023",

doi = "10.4171/EMSS/72",

language = "English",

volume = "10",

pages = "243--305",

journal = "EMS Surveys in Mathematical Sciences",

issn = "2308-2151",

publisher = "European Mathematical Society Publishing House",

}

RIS

TY - JOUR

T1 - String topology in three flavors

AU - Naef, Florian

AU - Rivera, Manuel

AU - Wahl, Nathalie

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