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I was in Stony Brook last week, visiting Moira Chas and Dennis Sullivan, and have been away from blogging for a while; this week I plan to write a few posts about some of the things I discussed with Moira and Dennis. This is an introductory post about the Goldman bracket, an extraordinary mathematical object made out of the combinatorics of immersed curves on surfaces. I don’t have anything original to say about this object, but for my own benefit I thought I would try to explain what it is, and why Goldman was interested in it.

In his study of symplectic structures on character varieties $\text{Hom}(\pi,G)/G$, where $\pi$ is the fundamental group of a closed oriented surface and $G$ is a Lie group satisfying certain (quite general) conditions, Bill Goldman discovered a remarkable Lie algebra structure on the free abelian group generated by conjugacy classes in $\pi$. Let $\hat{\pi}$ denote the set of homotopy classes of closed oriented curves on $S$, where $S$ is itself a compact oriented surface, and let $\mathbb{Z}\hat{\pi}$ denote the free abelian group with generating set $\hat{\pi}$. If $\alpha,\beta$ are immersed oriented closed curves which intersect transversely (i.e. in double points), define the formal sum

$[\alpha,\beta] = \sum_{p \in \alpha \cap \beta} \epsilon(p; \alpha,\beta) |\alpha_p\beta_p| \in \mathbb{Z}\hat{\pi}$

In this formula, $\alpha_p,\beta_p$ are $\alpha,\beta$ thought of as based loops at the point $p$, $\alpha_p\beta_p$ represents their product in $\pi_1(S,p)$, and $|\alpha_p\beta_p|$ represents the resulting conjugacy class in $\pi$. Moreover, $\epsilon(p;\alpha,\beta) = \pm 1$ is the oriented intersection number of $\alpha$ and $\beta$ at $p$.

This operation turns out to depend only on the free homotopy classes of $\alpha$ and $\beta$, and extends by linearity to a bilinear map $[\cdot,\cdot]:\mathbb{Z}\hat{\pi} \times \mathbb{Z}\hat{\pi} \to \mathbb{Z}\hat{\pi}$. Goldman shows that this bracket makes $\mathbb{Z}\hat{\pi}$ into a Lie algebra over $\mathbb{Z}$, and that there are natural Lie algebra homomorphisms from $\mathbb{Z}\hat{\pi}$ to the Lie algebra of functions on $\text{Hom}(\pi,G)/G$ with its Poisson bracket.

The connection with character varieties can be summarized as follows. Let $f:G \to \mathbb{R}$ be a (smooth) class function (i.e. a function which is constant on conjugacy classes) on a Lie group $G$. Define the variation function $F:G \to \mathfrak{g}$ by the formula

$\langle F(A),X\rangle = \frac {d}{dt}|_{t=0} f(A\text{exp}{tX})$

where $\langle \cdot,\cdot\rangle$ is some (fixed) $\text{Ad}$-invariant orthogonal structure on the Lie algebra $\mathfrak{g}$ (for example, if $G$ is reductive (eg if $G$ is semisimple), one can take $\langle X,Y\rangle = \text{tr}(XY)$). The tangent space to the character variety $\text{Hom}(\pi,G)/G$ at $\phi$ is the first cohomology group of $\pi$ with coefficients in $\mathfrak{g}$, thought of as a $G$ module with the $\text{Ad}$ action, and then as a $\pi$ module by the representation $\phi$. Cup product and the pairing $\langle\cdot,\cdot\rangle$ determine a pairing

$H^1(\pi,\mathfrak{g})\times H^1(\pi,\mathfrak{g}) \to H^2(\pi,\mathbb{R}) = \mathbb{R}$

where the last equality uses the fact that $\pi$ is a closed surface group; this pairing defines the symplectic structure on $\text{Hom}(\pi,G)/G$.

Every element $\alpha \in \pi$ determines a function $f_\alpha:\text{Hom}(\pi,G)/G \to \mathbb{R}$ by sending a (conjugacy class of) representation $[\phi]$ to $f(\phi(\alpha))$. Note that $f_\alpha$ only depends on the conjugacy class of $\alpha$ in $\pi$. It is natural to ask: what is the Hamiltonian flow on $\text{Hom}(\pi,G)/G$ generated by the function $f_\alpha$? It turns out that when $\alpha$ is a simple closed curve, it is very easy to describe this Hamiltonian flow. If $\alpha$ is nonseparating, then define a flow $\psi_t$ by $\psi_t\phi(\gamma)=\phi(\gamma)$ when $\gamma$ is represented by a curve disjoint from $\alpha$, and $\psi_t\phi(\gamma)= \text{exp} tF_\alpha(\phi)\phi(\gamma)$ if $\gamma$ intersects $\alpha$ exactly once with a positive orientation (there is a similar formula when $\alpha$ is separating). In other words, the representation is constant on the fundamental group of the surface “cut open” along the curve $\alpha$, and only deforms in the way the two conjugacy classes of $\alpha$ in the cut open surface are identified in $\pi$.

In the important motivating case that $G = \text{PSL}(2,\mathbb{R})$, so that one component of $\text{Hom}(\pi,G)/G$ is the Teichmüller space of hyperbolic structures on the surface $S$, one can take $f = 2\cosh^{-1}\text{tr/2}$, and then $f_\alpha$ is just the length of the geodesic in the free homotopy class of $\alpha$, in the hyperbolic structure on $S$ associated to a representation. In this case, the symplectic structure on the character variety restricts to the Weil-Petersson symplectic structure on Teichmüller space, and the Hamiltonian flow associated to the length function $f_\alpha$ is a family of Fenchel-Nielsen twists, i.e. the deformations of the hyperbolic structure obtained by cutting along the geodesic $\alpha$, rotating through some angle, and regluing. This latter observation recovers a famous theorem of Wolpert, connected in an obvious way to his formula for the symplectic form $\omega = \sum dl_\alpha \wedge d\theta_\alpha$ where $\theta$ is angle and $l$ is length, and the sum is taken over a maximal system of disjoint essential simple curves $\alpha$ for the surface $S$.

The combinatorial nature of the Goldman bracket suggests that it might have applications in combinatorial group theory. Turaev discovered a Lie cobracket on $\mathbb{Z}\hat{\pi}$, and showed that together with the Goldman bracket, one obtains a Lie bialgebra. Motivated by Stallings’ reformulation of the Poincaré conjecture in terms of group theory, Turaev asked whether a free homotopy class contains a power of a simple curve if and only if the cobracket of the class is zero. The answer to this question is negative, as shown by Chas; on the other hand, Chas and Krongold showed that a class $\alpha$ is simple if and only if $[\alpha,\alpha^3]$ is zero. Nevertheless, the full geometric meaning of the Goldman bracket remains mysterious, and a topic worthy of investigation.