# nLab Goldman bracket

### Context

#### Topology

topology

algebraic topology

# Contents

## 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 ${H}_{0}\left(L\Sigma \right)$ 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 $\left[\gamma \right]$ for the corresponding isotopy class.

For $\left[{\gamma }_{1}\right]$ and $\left[{\gamma }_{2}\right]$ 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 $\left[\gamma \right]$ is defined by

$\left\{\left[{\gamma }_{1}\right],\left[{\gamma }_{2}\right]\right\}:=\sum _{p\in {\gamma }_{1}\cap {\gamma }_{2}}\mathrm{sgn}\left(p\right)\left[{\gamma }_{1}{*}_{p}{\gamma }_{2}\right]\phantom{\rule{thinmathspace}{0ex}},$\left\{ [\gamma_1], [\gamma_2] \right\} := \sum_{p \in \gamma_1 \cap \gamma_2} sgn(p) [\gamma_1 \ast_p \gamma_2] \,,

where $\mathrm{sgn}\left(p\right)$ 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

Created on May 28, 2011 11:19:18 by Urs Schreiber (82.113.99.46)