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A chain-level model for Chas-Sullivan products in Morse homology with differential graded coefficients

Robin Riegel

TL;DR

The paper develops a finite-dimensional, chain-level Morse-theoretic framework for Chas–Sullivan-type string topology operations with differential graded coefficients. It constructs a DG–Morse model $CS_{DG}$ for a base–fiber product on enriched Morse complexes $C_*(X,\mathcal{F})$, proves a DG K"unneth formula and a DG Pontryagin–Thom construction, and proves a DG Thom isomorphism in the fibration setting. It extends Morse theory to $\mathcal{A}_{\infty}$-modules, develops continuation maps and invariance results, and shows functoriality with respect to coefficients and fibrations. The DG K"unneth machinery yields cross products $K^{alg}$ and $K^{top}$ that relate products on bases to products on total spaces, providing a robust, chain-level realization of string topology operations and their behavior under morphisms of fibrations. Overall, the work unifies Chas–Sullivan-type products, fibration theory, and higher algebra (A-infinity) in a finite-dimensional Morse framework with strong functorial and homological guarantees.

Abstract

We use the framework of Morse theory with differential graded coefficients to study certain operations on the total space of a fibration. More particularly, we focus in this paper on a chain-level description of the Chas-Sullivan product on the homology of the free loop space of an oriented, closed and connected manifold. The idea of ''intersecting on the base'' and ''concatenating on the fiber'' are well-adapted to this framework. We also give a Morse theoretical description of other products that follow the same principle. For this purpose, we develop functorial properties with respect to the coefficient in terms of morphisms of A $\infty$ -modules and morphisms of fibrations. We also build a differential graded version of the K{ü}nneth formula and of the Pontryagin-Thom construction.

A chain-level model for Chas-Sullivan products in Morse homology with differential graded coefficients

TL;DR

The paper develops a finite-dimensional, chain-level Morse-theoretic framework for Chas–Sullivan-type string topology operations with differential graded coefficients. It constructs a DG–Morse model for a base–fiber product on enriched Morse complexes , proves a DG K"unneth formula and a DG Pontryagin–Thom construction, and proves a DG Thom isomorphism in the fibration setting. It extends Morse theory to -modules, develops continuation maps and invariance results, and shows functoriality with respect to coefficients and fibrations. The DG K"unneth machinery yields cross products and that relate products on bases to products on total spaces, providing a robust, chain-level realization of string topology operations and their behavior under morphisms of fibrations. Overall, the work unifies Chas–Sullivan-type products, fibration theory, and higher algebra (A-infinity) in a finite-dimensional Morse framework with strong functorial and homological guarantees.

Abstract

We use the framework of Morse theory with differential graded coefficients to study certain operations on the total space of a fibration. More particularly, we focus in this paper on a chain-level description of the Chas-Sullivan product on the homology of the free loop space of an oriented, closed and connected manifold. The idea of ''intersecting on the base'' and ''concatenating on the fiber'' are well-adapted to this framework. We also give a Morse theoretical description of other products that follow the same principle. For this purpose, we develop functorial properties with respect to the coefficient in terms of morphisms of A -modules and morphisms of fibrations. We also build a differential graded version of the K{ü}nneth formula and of the Pontryagin-Thom construction.

Paper Structure

This paper contains 49 sections, 65 theorems, 350 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Recollections on Morse homology with DG coefficients
  3. Morse homology with DG coefficients
  4. Twisting cocycles defined by DG Morse data
  5. Fibration Theorem
  6. Lifting functions
  7. Statement
  8. Tools for the proof
  9. Direct and shriek maps
  10. Thom isomorphism in DG Morse theory. Proof of Theorem D
  11. Morse homology with coefficients in an Ai-module
  12. Ai-modules over C(OmegaX)
  13. Complex with coefficients in an Ai-module
  14. Ai-Morse toolset
  15. Filtration and spectral sequence
  16. ...and 34 more sections

Key Result

Theorem A

Let $F \hookrightarrow E \overset{\pi}{\to} X$ be a fibration and denote $\mathcal{F}=C_*(F)$ with the DG-module structure over $C_*(\Omega X)$ induced by a transitive lifting function $\Phi : E \space _{\pi}\!\times_{\textup{ev}_0} \mathcal{P} X \to E$ associated to the fibration. Let $m : E\ _{\pi The following properties hold : Moreover, this product corresponds in homology, via the Fibration

Figures (1)

  • Figure 1: Example of a Barraud-Cornea twisting cocycle on $\mathbb{S}^2$.

Theorems & Definitions (95)

  • Theorem A: \ref{['thm : A']}
  • Theorem B: Theorem \ref{['thm : morphisme induit commute avec iso']}
  • Theorem C: Theorem \ref{['thm : Künneth formula']}
  • Theorem D: Theorem \ref{['thm : DG Thom iso']}
  • Theorem E
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 85 more