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The Goresky-Hingston Coproduct in Morse Homology with DG Coefficients

Jonathan Clivio

TL;DR

This work develops a concrete Morse-theoretic model for the Goresky-Hingston coproduct on the free loop space with real coefficients by realizing the coproduct as a morphism of A_infty-modules over C_*(\Omega M). Central to the approach is the quasi-isomorphism C_*(\Lambda M) \simeq C_*(M, C_*(\Omega M)) and the construction of higher A_infty-morphisms \boldsymbol{\nu} whose first level recovers the coproduct on the based loop space. The authors introduce transitive diffeomorphism lifts and coherent chain homotopies, paired with invariant Thom classes via Mathai-Quillen theory, to obtain a robust algebraic framework that transfers GH coproduct data from \Omega M to the total loop space. A major payoff is the explicit computation of the GH coproduct for manifolds M = S^n/G and the description of its action on the Leray-Serre spectral sequence, including invariance properties under degree-1 maps. The results provide both concrete computable formulas and a structural, homotopy-coherent perspective on string topology operations in the Morse-theoretic setting, with implications for understanding coproducts beyond integer coefficients and for analyzing maps between manifolds.

Abstract

We describe the Goresky-Hingston coproduct on the free loop space with real coefficients via the quasi-isomorphism $C_*(ΛM)\simeq C_*(M,C_*(ΩM))$. This lets us describe the coproduct on the Leray-Serre spectral sequence as the diagonal on the manifold and the coproduct on the based loop space. With this description, we compute the coproduct with real coefficients for manifolds of the form $M=S^n/G$. The appendix contains results on inverting morphisms of $\mathcal{A}_\infty$-modules that are required for our computations.

The Goresky-Hingston Coproduct in Morse Homology with DG Coefficients

TL;DR

This work develops a concrete Morse-theoretic model for the Goresky-Hingston coproduct on the free loop space with real coefficients by realizing the coproduct as a morphism of A_infty-modules over C_*(\Omega M). Central to the approach is the quasi-isomorphism C_*(\Lambda M) \simeq C_*(M, C_*(\Omega M)) and the construction of higher A_infty-morphisms \boldsymbol{\nu} whose first level recovers the coproduct on the based loop space. The authors introduce transitive diffeomorphism lifts and coherent chain homotopies, paired with invariant Thom classes via Mathai-Quillen theory, to obtain a robust algebraic framework that transfers GH coproduct data from \Omega M to the total loop space. A major payoff is the explicit computation of the GH coproduct for manifolds M = S^n/G and the description of its action on the Leray-Serre spectral sequence, including invariance properties under degree-1 maps. The results provide both concrete computable formulas and a structural, homotopy-coherent perspective on string topology operations in the Morse-theoretic setting, with implications for understanding coproducts beyond integer coefficients and for analyzing maps between manifolds.

Abstract

We describe the Goresky-Hingston coproduct on the free loop space with real coefficients via the quasi-isomorphism . This lets us describe the coproduct on the Leray-Serre spectral sequence as the diagonal on the manifold and the coproduct on the based loop space. With this description, we compute the coproduct with real coefficients for manifolds of the form . The appendix contains results on inverting morphisms of -modules that are required for our computations.

Paper Structure

This paper contains 34 sections, 44 theorems, 293 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The Proof Strategy
  3. Background
  4. Morse Homology with DG Coefficients
  5. Data and Choices
  6. Construction
  7. Maps
  8. The Goresky-Hingston Coproduct
  9. Based Loop Space
  10. Sign Conventions
  11. Transitive Diffeomorphism Lifts
  12. Definition
  13. Construction
  14. Goresky-Hingston Coproduct as $\mathcal{A}_\infty$-Morphism
  15. Rewriting the Cap Croduct
  16. ...and 19 more sections

Key Result

Theorem A

Let $(M,x_0)$ be a smooth, pointed manifold of dimension $n$. There exists a morphism of $\mathcal{A}_\infty$-modules over $C_*(\Omega M)$ where $\nu_1$ computes the Goresky-Hingston coproduct on the based loop space. Moreover for the induced map $\widetilde{\nu}$, it holds that computes the Goresky-Hingston coproduct on homology of the free loop space.

Figures (10)

  • Figure 1: The vector field $v_{\lambda,t}$ is defined on the ball $B_{2\varepsilon}(\lambda(t))$ and points in the direction of $\dot{\lambda}(t)$.
  • Figure 2: The vector field $w_{\lambda,t}$ is defined on the ball $B_{2\varepsilon}(\lambda(t))$ and points inwards.
  • Figure 3: The vector field $V_\lambda$ is a linear combination of $v_{\lambda,t}(x)$ and $w_{\lambda,t}(x)$. In particular, it is zero for $x$ outside $B_{2\varepsilon}(\lambda(t))$ and $v_{\lambda,t}(x)$ on $B_{\frac{\varepsilon}{4}}(\gamma(t))$. The $w_{\lambda,t}(x)$-term is such that $V_\lambda$ moves the points with $d_M(x,\lambda(t))=\varepsilon$ closer to $\lambda(t)$ and the points with $d_M(x,\lambda(t))=\frac{\varepsilon}{2}$ further away from $\lambda(t)$.
  • Figure 4: A cycle that cuts out a fifth of the $V_{x_0}$.
  • Figure 5: The map $\varphi_{k_1,\dots,k_l}$.
  • ...and 5 more figures

Theorems & Definitions (115)

  • Theorem A: Theorem \ref{['thm:model of GH']}
  • Theorem B: Corollary \ref{['cor:coprod spec seq']}
  • Theorem C: Theorem \ref{['thm:GH on Sn/G']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Corollary 2.5
  • proof
  • Definition 2.6
  • ...and 105 more