# 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 $H_0(L \Sigma)$ 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

• 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

Revised on June 19, 2013 07:08:20 by Toby Bartels (98.23.135.113)