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 ${\gamma }_1{*}_p{\gamma }_2$ 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

$$\{[{\gamma }_1],[{\gamma }_2]\}:=\sum _{p\in {\gamma }_1\cap {\gamma }_2}\mathrm{sgn}(p)[{\gamma }_1{*}_p{\gamma }_2],$$\left\{ [\gamma_1], [\gamma_2] \right\} := \sum_{p \in \gamma_1 \cap \gamma_2} sgn(p) [\gamma_1 \ast_p \gamma_2] \,,

where $\mathrm{sgn}(p)$ 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

• W. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations , Invent. Math. (1986), no. 85, 263302.

The relation to string topology is due to

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