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String topology in three flavours

Florian Naef, Manuel Rivera, Nathalie Wahl

TL;DR

String topology studies algebraic structures on the free loop space of a manifold, focusing on the Chas–Sullivan loop product and the Goresky–Hingston loop coproduct. The paper develops and compares geometric ( Thom–Pontryagin) and algebraic (Hochschild/Tate–Hochschild) formulations of these operations, showing that together they form a unified algebra on the Tate–Hochschild complex of a dg Frobenius model; it also establishes an equivalence between geometric and algebraic descriptions in the simply connected, real-coefficient setting via configuration-space models. Lens-space computations demonstrate the coproduct’s discriminatory power and reveal not-fully-invariant behavior under homotopy, while the algebraic side uncovers a rich structure (cyclic $A_ty$-algebra, BV-operator, and Manin-triple interplay) that encapsulates endomorphisms in the singularity category. The work thereby ties Thom–Pontryagin geometry, Hochschild theory, and Fulton–MacPherson configuration-space models into a coherent framework for rational string topology, highlighting invariance properties and providing concrete computational and structural insights. Overall, it advances a deep, model-independent understanding of how loop-based operations encode manifold topology through both geometric and algebraic lenses and clarifies when and how these perspectives agree or diverge.

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.

String topology in three flavours

TL;DR

String topology studies algebraic structures on the free loop space of a manifold, focusing on the Chas–Sullivan loop product and the Goresky–Hingston loop coproduct. The paper develops and compares geometric ( Thom–Pontryagin) and algebraic (Hochschild/Tate–Hochschild) formulations of these operations, showing that together they form a unified algebra on the Tate–Hochschild complex of a dg Frobenius model; it also establishes an equivalence between geometric and algebraic descriptions in the simply connected, real-coefficient setting via configuration-space models. Lens-space computations demonstrate the coproduct’s discriminatory power and reveal not-fully-invariant behavior under homotopy, while the algebraic side uncovers a rich structure (cyclic -algebra, BV-operator, and Manin-triple interplay) that encapsulates endomorphisms in the singularity category. The work thereby ties Thom–Pontryagin geometry, Hochschild theory, and Fulton–MacPherson configuration-space models into a coherent framework for rational string topology, highlighting invariance properties and providing concrete computational and structural insights. Overall, it advances a deep, model-independent understanding of how loop-based operations encode manifold topology through both geometric and algebraic lenses and clarifies when and how these perspectives agree or diverge.

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.
Paper Structure (25 sections, 34 theorems, 147 equations, 2 figures)

This paper contains 25 sections, 34 theorems, 147 equations, 2 figures.

Table of Contents

  1. Introduction
  2. String topology via geometric intersection
  3. The intersection product as a Thom-Pontrjagin construction
  4. Definition of the product and coproduct as lifts of the intersection product
  5. Computation via geometric intersections
  6. Homotopy invariance
  7. The good and the bad coproduct
  8. String topology via Hochschild complexes
  9. Differential graded algebras
  10. Differential graded Frobenius algebras
  11. Hochschild chains and cochains
  12. Tate-Hochschild complex
  13. Two operations on Hochschild complexes
  14. Cyclic $A_{\infty}$-algebra on the Tate-Hochschild complex
  15. Final remarks
  16. ...and 10 more sections

Key Result

Theorem 1

The loop coproduct distinguishes non-homeomorphic 3-dimensional lens spaces.

Figures (2)

  • Figure 1: The retraction maps $R_{\mathrm{CS}}$ and $R_{\mathrm{GH}}$.
  • Figure :

Theorems & Definitions (73)

  • Theorem 1: Theorem \ref{['thm:lens']}
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3: Lifting the coproduct to a non-relative operation
  • Proposition 2.4
  • Proposition 2.5
  • ...and 63 more