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.
1
Trinity College, Dublin, Ireland
2
Purdue University, West Lafayette, USA
3
University of Copenhagen, Denmark
Florian Naef; Manuel Rivera; Nathalie Wahl. String topology in three flavors. EMS surveys in mathematical sciences, Tome 10 (2023) no. 2, pp. 243-305. doi: 10.4171/emss/72
@article{10_4171_emss_72,
author = {Florian Naef and Manuel Rivera and Nathalie Wahl},
title = {String topology in three flavors},
journal = {EMS surveys in mathematical sciences},
pages = {243--305},
year = {2023},
volume = {10},
number = {2},
doi = {10.4171/emss/72},
url = {http://geodesic.mathdoc.fr/articles/10.4171/emss/72/}
}
TY - JOUR
AU - Florian Naef
AU - Manuel Rivera
AU - Nathalie Wahl
TI - String topology in three flavors
JO - EMS surveys in mathematical sciences
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