Topics covered
 Goldman bracket, definition and Jacobi
identity (the Stone Alge4bra or a pair of pairs of points occuring
twice), examples.
 One to one correspondance between free homotopy
classes of oriented closed curves on a space X and conjugacy classes the
fundamamental group of X.
 The number of terms of the Goldman bracket of two classes is a
lower bound of their intersection number.
 "Following" intersection and selfintersection
point of curves on a surface along a homotopy.
 Homework: The Goldman Lie algebra of curves in the torus.
 Equivariant homology of the free loop space of a
free manifold (in degree 0 and 1) (equivariant with respect to the circle action rotating the domain)
 Homework: Let a and b denote the based homotopy
classes corresponding to the boundary components of the pair of
pant, oriented so that aB has selfintersection one. Consider the fundamental group of the
pairs of pants with generators a, b. Compute the brackets, [aB, Ab]
[aab, aB], [AAB, aB].
 The String Topology Lie algebra in the equivariant
homology of the free loop space of a manifold (We will work out
first on 3manifolds and degree 0 and 1 equivariant homology).
 Classification of three manifolds.
 The String
topology Lie algebra on a product of a surface with a circle. The String
topology Lie algebra on Seifert fibered spaces.
 Several examples of 1cycles in the equivariant
homology of the free loop space of a free manifold.

Brackets on
the zeroeth and first equivariant homology groups of the free loop
space of three manifolds.
Tentative List of Topics and References
Below is a preliminary list of topics, not necessarily in the order
they will be discussed.
The Lie bialgebra of curves on surfaces
 Definition and examples. Goldman.
 The Goldman Lie algebra of curves in the torus.
 The Goldman Lie algebra of curves in
a surface with boundary. Chas
 Relationship between the Goldman Lie algebra and
formal symplectic geometry. Kawazumi Kuno
 Digression: Selfintersections are Gaussian. ChasLalley
 The Goldman Lie algebra characterizes homeomorphisms. Gadgil,
 Intersection of and selfintersection number of curves
and relation to the Goldman Lie algebra. Chas, ChasKrongold
 The Lie bialgebra of curves on surfaces. Turaev
 Turaev's coalgebra and selfintersection of
curves. ChasKrongold, Le Donne.
 The center of the Goldman Lie algebra Etingoff,
Goldman Lie algebra for Fuchsian groups
 Definition and examples.
 The Goldman Lie algebra on the Modular Surface.
 Intersection of geodesics on the Modular surface.
String topology ChasSullivan
 Equivariant homology (of the free loop space) .
 A crash course on basic three manifold theory.
Hatcher's notes. Also, the first section of the beautifully
written Milnor's article. Finally, Brin's notes on Seifert fibered
spaces
 Lie algebra on the free loop space of three
manifolds. Definition and examples.
 The Lie algebra and the intersection number.
 The BV algebra structure on the homology of based
loop space of a manifold.
 Basu
 SullivanSullivan
 Detecting hyperbolicity of three manifolds
with the loop product. Abbaspour .
AndersenMattesReshetikhin algebra of chord diagrams Ellegaard
Andersen, Mattes, Reshetikhin and Cahn's
generalization's of Turaev's cobracket Cahn
