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String topology via the coHochschild complex and local intersections

Manuel Rivera, Alex Takeda

TL;DR

This work develops a tractable, local algebraic model for string topology operations by encoding the loop space through the coHochschild complex of a suitably enriched coalgebra derived from a simplicial complex with a local pairing. Central to the construction is a local homotopy pairing $\alpha$ that lifts intersection data to the level of a coalgebra, enabling explicit chain-level definitions of the loop product $\mu_{\alpha}$ and loop coproduct $\lambda_{\alpha}$ on $\mathop{\mathcal{coCH}}_*(C)$. The authors prove that, for fine triangulations of a smooth oriented closed manifold $M$, these algebraic operations coincide, up to chain homotopy, with the geometric Chas-Sullivan product and Goresky-Hingston coproduct, and extend to non-simply connected spaces and arbitrary coefficient rings. The locality framework, together with acyclic-model arguments, provides a robust path to generalizing string topology operations to homology manifolds and beyond, connecting combinatorial chain models with classical loop-space invariants. This work lays groundwork toward a full homotopy-invariant algebraic structure (e.g., an $IBL_{\infty}$-algebra) on loop spaces, driven by locality and coalgebraic analogues of pre-Calabi–Yau structures.

Abstract

We construct an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct in string topology. The construction takes as its initial input a simplicial complex equipped with a local pairing on its simplicial chains, for instance, a homology manifold with its local intersection pairing. We define the two string topology operations on the coHochschild complex of a suitable coalgebra of chains, making use of local higher homotopies that control the compatibility of the pairing with the diagonal approximation coproduct. In the case of a closed oriented smooth manifold, we prove that our algebraic operations coincide, up to chain homotopy, with their geometric counterparts. The local nature of our constructions allows for arguments based on the method of acyclic models.

String topology via the coHochschild complex and local intersections

TL;DR

This work develops a tractable, local algebraic model for string topology operations by encoding the loop space through the coHochschild complex of a suitably enriched coalgebra derived from a simplicial complex with a local pairing. Central to the construction is a local homotopy pairing that lifts intersection data to the level of a coalgebra, enabling explicit chain-level definitions of the loop product and loop coproduct on . The authors prove that, for fine triangulations of a smooth oriented closed manifold , these algebraic operations coincide, up to chain homotopy, with the geometric Chas-Sullivan product and Goresky-Hingston coproduct, and extend to non-simply connected spaces and arbitrary coefficient rings. The locality framework, together with acyclic-model arguments, provides a robust path to generalizing string topology operations to homology manifolds and beyond, connecting combinatorial chain models with classical loop-space invariants. This work lays groundwork toward a full homotopy-invariant algebraic structure (e.g., an -algebra) on loop spaces, driven by locality and coalgebraic analogues of pre-Calabi–Yau structures.

Abstract

We construct an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct in string topology. The construction takes as its initial input a simplicial complex equipped with a local pairing on its simplicial chains, for instance, a homology manifold with its local intersection pairing. We define the two string topology operations on the coHochschild complex of a suitable coalgebra of chains, making use of local higher homotopies that control the compatibility of the pairing with the diagonal approximation coproduct. In the case of a closed oriented smooth manifold, we prove that our algebraic operations coincide, up to chain homotopy, with their geometric counterparts. The local nature of our constructions allows for arguments based on the method of acyclic models.

Paper Structure

This paper contains 23 sections, 31 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Modeling the free loop space
  3. Categorical coalgebras
  4. Normalized chains
  5. Cobar construction
  6. A family of paths associated to an open necklace
  7. CoHochschild complex
  8. A family of free loops associated to a closed necklace
  9. Inverting edges
  10. Constant loops
  11. Fineness and locality
  12. Local pairings and controlled topology
  13. Simplicial complexes with local pairing
  14. Controlled Poincaré duality and homology manifolds
  15. Homotopy pairings on categorical coalgebras
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $X$ be a simplicial set determined by a coherent ordering of the simplices of a sufficiently fine simplicial complex $K$. Any degree $-n$ local pairing on the chains of $K$ lifts to a local homotopy pairing with cohomology class uniquely determined by $[\theta]\in H^n(K \times K)$. In particular, the intersection pairing of a homology manifold can be lifted to a local homotopy pairing on its

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 58 more