---
title: String Cobracket in Surface Topology
url: https://www.emergentmind.com/topics/string-cobracket
type: topic
---

# String Cobracket in Surface Topology

The string cobracket is an operation that records how a loop splits at its self-intersections. On an oriented surface, Turaev’s cobracket is defined on the free module generated by free homotopy classes of loops and sends a generic immersed loop to an antisymmetrized tensor sum of the two loops obtained by cutting at each transverse double point; together with the Goldman bracket it yields an involutive Lie bialgebra [1904.06686]. In string topology, the Goresky–Hingston coproduct induces a string cobracket on $S^1$-equivariant homology of the free loop space, and for closed surfaces this string cobracket is the negative of the Turaev cobracket [2510.05997].

## 1. Surface definition and geometric meaning

Let $S$ be a connected oriented surface, $\pi=\pi_1(S)$, and $\lvert K\pi\rvert=K(A)$ the free $K$-module on free homotopy classes $A\cong\pi/{\rm conj}$. If $\alpha\colon S^1\to S$ is a generic immersion with transverse double points, then the Turaev cobracket is the map
\[
\delta\;:\;\lvert K\pi\rvert\;\longrightarrow\;
\lvert K\pi\rvert\;\otimes\;\lvert K\pi\rvert
\]
given by
\[
\delta\bigl\lvert\alpha\bigr\rvert
\;=\;
\sum_{(t_1,t_2)\in D_\alpha}
\varepsilon\bigl(\alpha(t_1),\alpha(t_2)\bigr)\,
\bigl\lvert \alpha_{[t_1,t_2]}\bigr\rvert
\;\otimes\;
\bigl\lvert \alpha_{[t_2,t_1]}\bigr\rvert,
\]
where
\[
D_\alpha
=\bigl\{(t_1,t_2)\in S^1\times S^1\;;\;\alpha(t_1)=\alpha(t_2),\;t_1\neq t_2\bigr\},
\]
$\varepsilon(\alpha(t_1),\alpha(t_2))\in\{\pm1\}$ is the local intersection sign, and $\alpha_{[t_1,t_2]}$, $\alpha_{[t_2,t_1]}$ are the two arcs completed to loops [1904.06686]. In the equivalent geometric formulation used for surfaces with boundary,
\[
\delta([\,\gamma\,])
=\sum_{p\in I(\gamma)}
\bigl([\beta_{1,p}]\otimes[\beta_{2,p}]
-[\beta_{2,p}]\otimes[\beta_{1,p}]\bigr),
\]
where $I(\gamma)$ is the set of self-intersection points of a generic immersed loop $\gamma$ and the two local branches are ordered so that the pair of tangent vectors agrees with the orientation of the surface [1009.2620].

This definition makes precise the statement that the cobracket “measures self-intersection of a loop on a surface” [1904.06686]. If $C\subset S$ is embedded with no self-intersection, then one can choose a representative with no transverse double points, and hence
\[
\delta(\lvert C\rvert)=0
\]
[1904.06686]. The vanishing on simple loops is therefore immediate from the definition, whereas the converse is more subtle.

## 2. Lie-coalgebraic structure and the simplicity problem

Turaev’s cobracket is coantisymmetric, satisfies the co-Jacobi identity, and is compatible with the Goldman bracket. More precisely, if
\[
\{\,,\,\}\colon \lvert K\pi\rvert\otimes \lvert K\pi\rvert\to \lvert K\pi\rvert
\]
denotes the Goldman bracket, then $\delta$ satisfies
\[
\tau\circ\delta=-\delta,
\]
the co-Jacobi identity, and the Lie-bialgebra compatibility
\[
\delta\bigl\{\lvert\alpha\rvert,\lvert\beta\rvert\}
=\;
{\rm ad}_{\lvert\alpha\rvert}\bigl(\delta\lvert\beta\rvert\bigr)\;-\;
{\rm ad}_{\lvert\beta\rvert}\bigl(\delta\lvert\alpha\rvert\bigr).
\]
Accordingly, $\bigl(\lvert K\pi\rvert,\{\,,\,\},\delta\bigr)$ is a Lie bialgebra in the sense of Drinfel’d, and Chas further showed that this Lie bialgebra is involutive [1904.06686].

A central problem is whether vanishing of the cobracket detects embedded curves. For a non-power conjugacy class $x$, the statement
\[
\delta(x^k)=0 \quad\text{if and only if}\quad x \text{ is represented by an embedded simple closed curve}
\]
is denoted Turaev$(k)$ [1009.2620]. The simplest version is false: computer computations show counterexamples to Turaev$(1)$ on every surface of negative Euler characteristic except the pair of pants [1009.2620]. By contrast, the main theorem of Chas–Krongold proves Turaev$(k)$ for all $k\ge 3$ on surfaces with boundary. More precisely, for any nonpower $x$ and any integer $p\ge 4$,
\[
\|\delta(x^p)\|_{\mathrm{Manh}}
=p^2\,s(x),
\]
where $s(x)$ is the minimal self-intersection number of $x$ [1009.2620].

This establishes an important correction to a common overstatement. Vanishing of the string cobracket is not, by itself, an intrinsic characterization of simplicity at the level $k=1$; however, sufficiently high powers do detect simplicity on surfaces with boundary [1009.2620].

## 3. Framed, completed, and Hodge-theoretic formulations

The unframed cobracket changes by monogon moves, so a framing yields a homotopy-invariant refinement. If $f$ is a trivialization of the tangent bundle $TS$ and $\mathrm{rot}_f(\alpha)\in\mathbb Z$ is the rotation number of an immersed loop $\alpha$, then the framed cobracket is
\[
\delta^f\lvert\alpha\rvert
=
\delta\lvert\alpha\rvert
+\mathrm{rot}_f(\alpha)\,\bigl(1\otimes\lvert\alpha\rvert-\lvert\alpha\rvert\otimes 1\bigr).
\]
The pair $\bigl(\lvert K\pi\rvert,\{\,,\,\},\delta^f\bigr)$ is again an involutive Lie bialgebra, and $\delta^f$ is fully invariant under free homotopy of $\alpha$ [1904.06686].

After completing in the $I$-adic topology, the framed Turaev cobracket acquires additional structure. For a smooth affine curve $X$ over $\mathbb C$ with an algebraic framing $\xi$, the completed cobracket
\[
\delta_\xi\;:\; \bigl(\lambda(X)^\wedge\,,\,W\,,\,F\bigr)\;\otimes(-1)
\;\longrightarrow\;
\lambda(X)^\wedge\;\widehat\otimes\;\lambda(X)^\wedge
\]
is a morphism of pro–mixed Hodge structure [1807.09209]. Equivalently, $\lambda(X)^\wedge\otimes(1)$ is a Lie coalgebra in the category of pro–mixed Hodge structures, and together with the Goldman bracket it forms an involutive Lie bialgebra in MHS [1807.09209].

The same circle of ideas has a homological formulation. Hain’s construction uses the real-oriented blow-up $\widetilde{C}_2(M)$ of $M\times M$ along the diagonal and produces a relative $2$-cycle attached to an immersed loop; capping with the framing class reproduces exactly the framed cobracket $\delta^f$ [1904.06686]. This suggests a robust interpretation of the string cobracket as an intersection-theoretic operation encoded in configuration-space geometry.

## 4. Tensorial, combinatorial, and divergence descriptions

For genus-$0$ compact surfaces, the cobracket admits a tensorial description through the standard group-like expansion. If $S=S_{0,n+1}$ and $\theta_{\rm std}$ is characterized by
\[
\theta_{\rm std}(Y_k)=\exp(x_k)\qquad (k=1,\dots,n),
\]
then extending scalars and passing through $\theta_{\rm std}$ yields a continuous map
\[
\delta^{\mathrm{std}} : N(T) \to N(T)\otimes N(T),
\]
whose explicit formula contains Bernoulli numbers [1506.03174]. The power series
\[
s(z)=\frac1{e^{-z}-1}+\frac1{z}
=-\frac12-\frac z{12}+\frac{z^3}{720}-\cdots
\]
enters both the tensorial formula for the coaction and the description of the homotopy-intersection pairing, forcing Bernoulli numbers onto the final $\delta$-formula [1506.03174]. In particular, the leading part recovers the classical “necklace” cobracket of Schedler, while the full series refines it by higher self-intersection data [1506.03174].

A complementary algebraic description uses non-commutative divergence. For a formally smooth associative algebra $A$ with a flat connection
\[
\nabla\!:\Omega^1A\to \Omega^1(A^e)\otimes_{A^e}\Omega^1A,
\]
the divergence map is
\[
{\rm Div}^\nabla(f)=
{\rm Tr}\bigl(L_f-(i_{\tilde f}\otimes{\rm id}_{\Omega^1A})\circ\nabla\bigr).
\]
If $\nabla$ is flat, then ${\rm Div}^\nabla$ is a Lie-algebra $1$-cocycle [2403.16566]. For a compact oriented surface with boundary and a free generating system $\mathcal C$, one has a canonical flat connection $\nabla_{\mathcal C}$ and a based Goldman map
\[
\sigma:|\mathbb K\pi|\to {\rm Der}_{\mathbb K\langle\zeta_0\rangle}(\mathbb K\pi),
\]
with the identity
\[
\delta^{fr}= {\rm Div}^{\mathcal C}\circ \sigma
\]
for the rotation-free framing [2403.16566]. For closed surfaces, a homological connection on a perfect projective resolution of $\Omega^1(\mathbb K\pi)$ yields an analogous formula
\[
{\rm Div}^{\nabla'}\circ v=\delta,
\]
where $v:|\mathbb K\pi|\to HH^1(\mathbb K\pi)$ is Vaintrob’s map [2403.16566].

These descriptions show that the string cobracket is not merely a local cutting rule. It also admits tensorial, homological, and non-commutative algebraic realizations.

## 5. Refinements and extensions of the surface theory

A major refinement is Patricia Cahn’s operation $\mu$, defined on free homotopy classes of loops on an oriented surface by replacing each self-intersection with a single chord diagram rather than a pure smoothing. If $E$ is the free $\mathbb Z$-module on one-chord diagrams, then
\[
\mu(\alpha)
=
\sum_{(t_1,t_2)\in\mathcal{SI}_0(a)}
\Bigl([\,a^1_p\!\bullet_p a^2_p]-[\,a^2_p\!\bullet_p a^1_p]\Bigr)\in E,
\]
and there is a canonical smoothing projection
\[
Q:E\to \mathbb Z\langle\mathcal L\rangle\otimes \mathbb Z\langle\mathcal L\rangle
\]
such that
\[
\Delta = Q\circ \mu.
\]
Thus $\mu$ is a refinement of the classical cobracket [1004.0532]. It satisfies coskew-symmetry and a co-Jacobi-type identity, and it yields an exact formula for the minimal self-intersection number:
\[
m(\alpha)=(n-1)+\frac12\,t(\mu(\alpha)),
\]
when $\alpha=\beta^n$ with $\beta$ primitive [1004.0532]. In particular, $\mu(\alpha)=0$ if and only if $\alpha$ is a power of a simple class [1004.0532].

The same refinement has a virtual-string analogue. For a virtual string $\alpha$, Turaev’s virtual-string cobracket $\nu$ and the generalized operation $\mu$ satisfy
\[
\nu = S\circ \mu,
\]
where $S$ is the “smoothing-off the sign” map [1107.4718]. If the minimal representative of $[\alpha]$ realizes an $n$-fold power in $\pi_1$ of its supporting surface, then
\[
m([\alpha])\ge t(\mu([\alpha]))/2+(n-1).
\]
For $n=1$, this reduces to
\[
m([\alpha])\ge t(\mu([\alpha]))/2
\]
[1107.4718]. The paper also constructs a $5$-arrow virtual string $M$ with
\[
\rho([M])=4,\qquad t(\mu([M]))=10,\qquad m([M])=5,
\]
so that
\[
t(\mu([M]))/2=5>\rho([M])=4,
\]
showing that $\mu$ can be strictly stronger than both the virtual-string cobracket bound and Turaev’s based-matrix invariant [1107.4718].

A further extension is a family of maps $\delta_k$ parametrised by simple ribbon graphs with $k$ vertices. For $A=\mathbb Q[\pi_1\Sigma]$, $k=1$ recovers the framed Turaev cobracket, $k=2$ gives the double divergence, and for $A=T(W)$ the operations coincide with the ribbon-graph operations $L_k$ in $\mathrm{RGra}_1$ [2502.04806]. When the connection is flat, the odd $\delta_k$ define Chevalley–Eilenberg cocycles, and in the free associative case they recover the standard odd generators of $H^*(\mathfrak{gl}_n,\mathbb Q)$ [2502.04806].

## 6. String topology, surface comparison, and higher-dimensional limitations

In string topology, the string cobracket is defined on $S^1$-equivariant homology by combining the transfer map with the Goresky–Hingston coproduct. For a closed oriented $n$-manifold $M$, one defines
\[
\{\, - \,\}: H_i^{S^1}(LM)\to H_{i+2-n}^{S^1}(LM)\otimes H_{i+2-n}^{S^1}(LM)
\]
as the composite
\[
H_i^{S^1}(LM)\xrightarrow{\mu}H_{i+1}(LM)\to H_{i+1}(LM,M)\xrightarrow{\vee}
H_{i+2-n}(LM,M)^{\otimes 2}\xrightarrow{\eta\otimes\eta}
H_{i+2-n}^{S^1}(LM)^{\otimes 2}.
\]
For a closed surface, the only nontrivial case is $i=0$, and under the natural isomorphism
\[
H_0^{S^1}(LM)\cong V:=\mathrm{Span}_{\mathbb Q}\{\text{free-homotopy classes of loops}\}
\]
one obtains a degree-$0$ cobracket [2510.05997]. The main comparison theorem states that
\[
\{x\}=-\,\vee_T(x)\qquad (\forall\,x\in V),
\]
so on surfaces the string topology cobracket is the negative of the Turaev cobracket [2510.05997].

At chain level, Naef–Willwacher model the string-topology cobracket algebraically on cyclic chains and construct an $IBL_\infty$-structure such that the natural comparison map to the $S^1$-equivariant loop-space homology intertwines the Lie bialgebra structure on homology [1911.06202]. Their construction depends on the perturbative partition function of a Chern–Simons type topological field theory, and the one-loop part introduces terms in the cobracket that “do not depend only on the real homotopy type of $M$, but on its finer smooth structure” [1911.06202]. The later chain-level analysis via configuration-space integrals and Chen’s iterated integrals proves that homotopy transfer intertwines the involutive Lie bialgebra structures on homology [2607.03782].

The surface picture does not extend unchanged to higher-dimensional string topology. For lens spaces $L(9;1)$ and $L(9;4)$, which are homotopy equivalent but not homeomorphic, explicit computations show that for $k=1$ there are in total $11$ nonzero components, whereas for $k=4$ there are in total $14$ nonzero components; accordingly, “the string cobracket is not a homotopy invariant” [2605.10273]. The same work proves that there exists a closed manifold $M$—namely $L(9;4)$—for which the string bracket and the string cobracket do not satisfy Drinfeld compatibility, so $H_*^{S^1}(LM)$ does not carry an honest Lie bialgebra structure over $\mathbb Z$ in higher dimensions [2605.10273].

A plausible implication is that the phrase “string cobracket” has two closely related but not universally interchangeable meanings: on surfaces it refers to Turaev’s loop-splitting Lie cobracket, while in higher-dimensional string topology it denotes an $S^1$-equivariant operation whose homotopy invariance and Lie-bialgebraic behavior require additional hypotheses.

Source: https://www.emergentmind.com/topics/string-cobracket