nLab
Goldman bracket

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Contents

Idea

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

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

Definition

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

For [γ 1][\gamma_1] and [γ 2][\gamma_2] two such classes, one can always find differentiable representatives γ 1\gamma_1 and γ 2\gamma_2 that intersect - if they intersect at some point pp - transversally. Write γ 1* pγ 2\gamma_1 \ast_p \gamma_2 for the curve obtained by starting at the intersection point pp, traversing along γ 1\gamma_1 back to that point and then along γ 2\gamma_2.

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

{[γ 1],[γ 2]}:= pγ 1γ 2sgn(p)[γ 1* pγ 2], \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)sgn(p) is +1 if T pγ 1,T pγ 2T_p \gamma_1, T_p \gamma_2 is an oriented basis of the tangent space T pΣT_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)