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The Goresky-Hingston Coproduct on Based Loop Spaces

Jonathan Clivio

TL;DR

This work constructs a lift of the Goresky-Hingston coproduct to the based loop space via the avoiding-stick coproduct, enabling a tractable, non-relative chain-level model. It proves a $\,\pi_1(M,x_0)$-invariance result and a naturality formula for maps $f:M\to N$, allowing explicit computation of the coproduct for manifolds with finite fundamental groups and, in particular, for $M=S^n/G$. The authors derive concrete algebraic structures: $H_*(\Omega M)\cong \mathbb{Z}[G][x]$ with $|x|=n-1$ and a precise coproduct formula $\lor(gx^k)=\sum_{i+j=k-1}\sum_{h\in G} gh^{-1}x^i\otimes hx^j$, and show the corresponding Leray-Serre spectral sequence for the free loop space collapses, yielding a decomposed, computable description of $H_*(\Lambda M)$. The results provide a practical framework for string topology on quotients of spheres, connecting based- and free-loop coproducts via universal covers and group actions, with implications for the interplay between homotopy, coverings, and loop-space algebraic structures.

Abstract

We construct a lift of the Goresky-Hingston coproduct on the based loop space. Using this lift, we produce formulas for the interaction of the coproduct with the $π_1$-action and with maps of manifolds. These results lets us compute the coproduct on the based loop space for manifolds of the form $S^n/G$.

The Goresky-Hingston Coproduct on Based Loop Spaces

TL;DR

This work constructs a lift of the Goresky-Hingston coproduct to the based loop space via the avoiding-stick coproduct, enabling a tractable, non-relative chain-level model. It proves a -invariance result and a naturality formula for maps , allowing explicit computation of the coproduct for manifolds with finite fundamental groups and, in particular, for . The authors derive concrete algebraic structures: with and a precise coproduct formula , and show the corresponding Leray-Serre spectral sequence for the free loop space collapses, yielding a decomposed, computable description of . The results provide a practical framework for string topology on quotients of spheres, connecting based- and free-loop coproducts via universal covers and group actions, with implications for the interplay between homotopy, coverings, and loop-space algebraic structures.

Abstract

We construct a lift of the Goresky-Hingston coproduct on the based loop space. Using this lift, we produce formulas for the interaction of the coproduct with the -action and with maps of manifolds. These results lets us compute the coproduct on the based loop space for manifolds of the form .

Paper Structure

This paper contains 14 sections, 27 theorems, 141 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Avoiding-Stick Coproduct
  3. Various Coproducts
  4. The Trivial Coproduct
  5. The Goresky-Hingston Coproduct
  6. The Avoiding-Stick Definition
  7. Cutting at Other Points
  8. The Trivial Pre-Coproduct
  9. The Avoiding-Stick Pre-Coproduct
  10. Higher Dimensional Spheres
  11. Maps and the Coproduct
  12. Universal Covering
  13. Spheres Modulo a Group Action
  14. The Free Loop Space

Key Result

Theorem A

Assume that the dimension is $n>1$. The avoiding-stick coproduct $\lor$ factors through the fixed points $H_*(\Omega M\times \Omega M)^{\pi_1(M,x_0)}$ where $\pi_1(M,x_0)$ acts on $\Omega M\times \Omega M$ via

Figures (6)

  • Figure 1: The space $\Omega^\sigma M$ consists of all loops that start and end with the path $\sigma$.
  • Figure 2: In this picture, the $1$-parameter family $H$ is given by translation downwards. The blue loop $\gamma$ is in $\Omega^\sigma M$ and has a self-intersection at $t$. Its image $\Phi(\gamma)$ in yellow then is in $\Omega^{\widetilde{\sigma}}M$ and at $t$ it crosses $x_1=\phi^\varepsilon(x_0)$.
  • Figure 3: The graph of the function $\chi$.
  • Figure 4: The set up of Proposition \ref{['prop:a-s commuting with homotopy']} for $M=S^2$ where the $1$-parameter family is given by a flow from the north pole to the south pole. The path $\sigma_2$ is then given by a "bent version" of $\sigma_2$.
  • Figure 5: In the case $x_0=x_1$, the stick $\sigma_1$ is going to the right and the path $\tau$ and the $1$-parameter family is moving upwards in a local chart. Here we use the assumption that $n>1$.
  • ...and 1 more figures

Theorems & Definitions (65)

  • Theorem A: Corollary \ref{['cor:pi1 invariance']}
  • Theorem B: Corollary \ref{['cor:formula for f']}
  • Theorem C: Remark \ref{['rem:Omega X univ cov']} and Proposition \ref{['prop:GH via univ cov']}
  • Theorem D: Proposition \ref{['prop:Omega Sn/G']} and Corollary \ref{['cor:coprod Omega Sn/G']}
  • Theorem E: Corollary \ref{['cor:free loop space']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 55 more