CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
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
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.
The original definition is due to
The relation to string topology is due to