Introduction to Heegaard Floer Seminar, Fall 2015


The Heegaard Floer learning seminar met Fridays of Fall 2015.

 Schedule:

September 11: Handle decompositions and Heegaard diagrams- Lisa
September 18: Morse theory and Morse homology- Martin
September 25: Preliminary symplectic geometry- Alex
October 2: Symmetric products of surfaces- Andrew
October 9: Intro to Lagrangian Floer homology- Max.
October 16: Definition of Heegaard Floer homology- Allison.
October 30: Invariance of Heegaard Floer- Jonathan.
November 6: Combinatorial Heegaard Floer -Ahmad.
November 13: Flavors of Heegaard Floer- Allison and Andrew
November 20: Knot Heegaard Floer- Lisa.
November: Cobordism maps and surgery formulas- Kyle
Links are to talk notes by Andrew.



Resource List:

Lecture notes:

An Introduction to Heegaard Floer Homology- P. Ozsváth and Z. Szabó.
Gives background definitions and facts, introduces HF and gives examples of HF and HFK.

Lectures on Heegaard Floer Homology- P. Ozsváth and Z. Szabó.
A sequel to the previous lecture notes- surgery exact triangle is introduced and proved, discussion of maps induced by smooth cobordisms, and other applications.

An introduction to knot Floer homology- C. Manolescu.

Lecture Notes- E. Tweedy.
Lecture notes from a course taught at Rice by E. Tweedy.

Original Papers:

Holomorphic discs and topological invariants for closed three-manifolds.  P. Ozsváth and Z. Szabó.
Holomorphic discs and three-manifold invariants: Properties and applications. P. Ozsváth and Z. Szabó.
Discusses relationship between Euler characteristic of HF and Turaev torsion, relationship to Thurston norm, and surgery exact sequence.
Combinatorial Heegaard Floer homology and nice Heegaard diagrams.  P. Ozsváth,  A. Stipsicz,  Z. Szabó.
Gives a combinatorial description of (a variant of) HF and proves invariance combinatorially.
Holomorphic discs and knot invariants. P. Ozsváth and Z. Szabó.
Introduces knot Heegard Floer homology. (Done also independently by J. Rasmussen here.)

Holomorphic triangles and invariants for smooth four-manifolds. P. Ozsváth and Z. Szabó.
Introduces maps on HF associated to cobordisms and invariants associated to closed smooth 4-manifolds.
Absolutely Graded Floer homologies and intersection forms for four-manifolds with boundary. P. Ozsváth and Z. Szabó.
Gives obstructions for surgery on a knot being a lens space, reproves Donaldson's Intersection Theorem, among many other things.




Rough talk descriptions:
We'll add links to talk plans as they get written!


Morse theory and Morse homology.

Handle decompositions and Heegaard diagrams.
   
Preliminary symplectic stuff.

Symmetric products of surfaces.
 
Intro to Lagrangian Floer homology.

Definition of HF.

Invariance of HF.

Combinatorial HF:
Give a combinatorial definition of Heegaard Floer. What are the restrictions on the diagrams? Do some examples. When is this reasonable to compute?


Flavors of HF:
Talk about HF-hat, HF+, HF-, HF-infty. Give some examples. What's the difference in the information captured? What's the advantage of having all of these  definitions? What's different about proving invariance? Maybe talk about the surgery exact sequence.


Knot HF:
Give the definition of Knot Heegaard Floer homology, focusing primarily on the case of a knot in S^3. (You might want to ignore the discussion of multiple basepoints for simplicity.) Give some simple examples. Talk about HFK-hat as well as CFK-infty- what's the difference in the information captured? What are the important properties (relationship to Alexander polynomial, connected sum, reflection and orientation change, etc)?

Cobordism maps and surgery formulas: