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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.


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