Equivalently, the Goldman bracket on is a structure on the 0th homology of the free loop space of . It is in fact just the lowest degree of the string topology operations on . See there for more details.
For and two such classes, one can always find differentiable representatives and that intersect - if they intersect at some point - transversally. Write for the curve obtained by starting at the intersection point , traversing along back to that point and then along .
The Goldman bracket on the free abelian group on classes is defined by