nLab Goldman bracket

Idea

The Goldman bracket of a compact closed surface $\Sigma$ is a Lie algebra structure on the free abelian group generated from the isotopy classes of based loops in $\Sigma$.

Equivalently, the Goldman bracket on $\Sigma$ is a structure on the 0th homology $H_0(L \Sigma)$ of the free loop space of $\Sigma$. It is in fact just the lowest degree of the string topology operations on $\Sigma$. See there for more details.

Definition

Let $\Sigma$ be a compact closed and oriented surface (manifold of dimension 2). For $\gamma : S^1 \to \Sigma$ a continuous function from the based circle, write $[\gamma]$ for the corresponding isotopy class.

For $[\gamma_1]$ and $[\gamma_2]$ two such classes, one can always find differentiable representatives $\gamma_1$ and $\gamma_2$ that intersect - if they intersect at some point $p$ - transversally. Write $\gamma_1 \ast_p \gamma_2$ for the curve obtained by starting at the intersection point $p$, traversing along $\gamma_1$ back to that point and then along $\gamma_2$.

The Goldman bracket on the free abelian group on classes $[\gamma]$ is defined by

$$\left\{ [\gamma_1], [\gamma_2] \right\} := \sum_{p \in \gamma_1 \cap \gamma_2} sgn(p) [\gamma_1 \ast_p \gamma_2] \,,$$

where $sgn(p)$ is +1 if $T_p \gamma_1, T_p \gamma_2$ is an oriented basis of the tangent space $T_p \Sigma$, and -1 otherwise.

References

The original definition is due to

• William M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), no. 85, 263–302 doi

The relation to string topology is due to

• Moira Chas, Dennis Sullivan, String topology, Ann. Math. math.GT/9911159