nLab
Goldman bracket

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)