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String topology coproduct and Turaev cobracket on surfaces

Jana Hartenstein, Maximilian Stegemeyer

TL;DR

This work provides a complete computation of the string topology coproduct on the free loop space of closed orientable surfaces of genus $g\ge 2$ by encoding free loop classes as cyclic words in the fundamental group and describing an explicit algorithm that records transversal self-intersections to determine $\bigvee\Delta[\gamma]$. It establishes that the string cobracket on $H_*^{S^1}(LM)$ is the negative of the Turaev cobracket, clarifying sign conventions via a transfer‑map framework. The authors also analyze the homology of $LM$ for surfaces, give a detailed construction realizing the coproduct algorithm, and discuss higher‑dimensional extensions showing vanishing results for hyperbolic and certain 3‑manifolds, highlighting where the genus‑zero/low‑dimensional intuition does not carry over. These results provide both an explicit computational toolset and a sharper conceptual link between string topology and classical surface invariants such as the Turaev cobracket and Goldman bracket.

Abstract

The string topology coproduct on the homology of the free loop space of a closed manifold induces a string cobracket on $S^1$-equivariant homology. We give a complete computation of the string topology coproduct for surfaces of higher genus by describing an algorithm which computes the coproduct of a cyclic word in terms of generators of the fundamental group of the surface. We further show that the string cobracket is the negative of the Turaev cobracket.

String topology coproduct and Turaev cobracket on surfaces

TL;DR

This work provides a complete computation of the string topology coproduct on the free loop space of closed orientable surfaces of genus by encoding free loop classes as cyclic words in the fundamental group and describing an explicit algorithm that records transversal self-intersections to determine . It establishes that the string cobracket on is the negative of the Turaev cobracket, clarifying sign conventions via a transfer‑map framework. The authors also analyze the homology of for surfaces, give a detailed construction realizing the coproduct algorithm, and discuss higher‑dimensional extensions showing vanishing results for hyperbolic and certain 3‑manifolds, highlighting where the genus‑zero/low‑dimensional intuition does not carry over. These results provide both an explicit computational toolset and a sharper conceptual link between string topology and classical surface invariants such as the Turaev cobracket and Goldman bracket.

Abstract

The string topology coproduct on the homology of the free loop space of a closed manifold induces a string cobracket on -equivariant homology. We give a complete computation of the string topology coproduct for surfaces of higher genus by describing an algorithm which computes the coproduct of a cyclic word in terms of generators of the fundamental group of the surface. We further show that the string cobracket is the negative of the Turaev cobracket.

Paper Structure

This paper contains 6 sections, 13 theorems, 45 equations, 5 figures.

Table of Contents

  1. Introduction
  2. String topology operations
  3. The homology of the free loop space of surfaces
  4. Computing the coproduct
  5. String cobracket and Turaev Cobracket
  6. The coproduct for hyperbolic manifolds and $3$-manifolds

Key Result

Theorem 1

Let $M$ be a closed oriented surface of genus $g\geq 2$. Let $v = v_1 \ldots v_m$ be a cyclically reduced representative of a cyclic word in the alphabet $\{c_1,\ldots, c_{2g},c_1^-,\ldots ,c_{2g}^- \}$ and let $[\gamma]\in {H}_0(LM)$ be the induced free homotopy class of loops in $M$. There is an a

Figures (5)

  • Figure 1: Deformation of $\gamma^m$ in a tubular neighborhood of $\gamma$
  • Figure 2: The wedge of circles $\bigvee_{i=1}^4 {S}^1$ with cyclic order of the half-edges at the vertex and its fattening. The orientation of the boundary of the fattening is indicated. We also sketch the neighborhood $U_0$ of the basepoint.
  • Figure 3: Construction of the loop respresenting the word $c_1c_1c_3c_1^{-1} c_3$ for genus $g= 2$. The following cases of intersections appear: $\mu_2$ intersects $\delta_1$, this is case 1.c). Further, $\mu_5$ intersects $\delta_1$ and $\mu_2$, both are case 1.e) and $\mu_5$ intersects with $\mu_3$, this is case 1.c). Finally, $\delta_6$ intersects $\mu_2$, this is case 3.a) and $\delta_6$ and $\mu_4$ intersect giving an example of case 3.d).
  • Figure 4: Sketch of the twelve cases of intersections appearing in the algorithm. For each case we record the corresponding sign which is determined by Lemma \ref{['lemma_coproduct_signs']}. Note that in the cases 1.a) - 1.f) as well as 2.a) and 2.b) the part $f_k$ or $\overline{f}_k$ is run through first by the loop before $\mu_j$. In the cases 3.a) - 3.d) the part $\mu_k$ is run through first before $\overline{f}_m$. For cases 1.c) - 1.f) as well as 3.c) and 3.d) where the ends agree, we draw the original position of the end $e_*$ in order to point out where the intersections happen.
  • Figure 5: The representative $\gamma$ for the word $c_4 c_6 c_3 c_1^{-1} c_5^{-1} c_4$. The parts $\mu_3$ and $\mu_5$ intersect, see case 2.b) in the proof of Theorem \ref{['theorem_algorithm']} as well as $\mu_2$ and $\delta_7$ which is case 3.d).

Theorems & Definitions (31)

  • Theorem : Theorem \ref{['theorem_algorithm']}
  • Theorem : Theorem \ref{['theorem_turaev']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: HingstonWahl, Prop. 3.7
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • ...and 21 more