- Goldman bracket, definition and Jacobi
identity (the Stone Alge4bra or a pair of pairs of points occuring
- 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 self-intersection
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 self-intersection 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 3-manifolds 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 1-cycles in the equivariant
homology of the free loop space of a free manifold.
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
Goldman Lie algebra for Fuchsian groups
- 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: Self-intersections are Gaussian. Chas-Lalley
- The Goldman Lie algebra characterizes homeomorphisms. Gadgil,
- Intersection of and self-intersection number of curves
and relation to the Goldman Lie algebra. Chas, Chas-Krongold
- The Lie bialgebra of curves on surfaces. Turaev
- Turaev's coalgebra and self-intersection of
curves. Chas-Krongold, Le Donne.
- The center of the Goldman Lie algebra Etingoff,
String topology Chas-Sullivan
- Definition and examples.
- The Goldman Lie algebra on the Modular Surface.
- Intersection of geodesics on the Modular surface.
Andersen-Mattes-Reshetikhin algebra of chord diagrams Ellegaard
Andersen, Mattes, Reshetikhin and Cahn's
generalization's of Turaev's cobracket Cahn
- 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
- 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.
- Detecting hyperbolicity of three manifolds
with the loop product. Abbaspour .