Math 235: Topics in Manifold Theory

Combinatorial Heegaard Floer Homology

Fall 2011

  • Time and Place: MW 1-2:20 pm in MS 7608
  • Instructor: Ciprian Manolescu
  • E-mail:
  • Office: MS 6921
  • Office Hours: Wednesdays 10-11am and by appointment


Heegaard Floer theory, developed by Ozsvath and Szabo, is a powerful technique for studying the key objects in low-dimensional topology: knots and links in the three-sphere, 3- and 4-dimensional manifolds. In particular, it provides answers to the following questions: Given a knot in space, how can we tell if it is the unknot? Given a two-dimensional homology class in a three-manifold, what is the minimal complexity of a surface representing that class? How can one distinguish 4-manifolds that are homeomorphic but not diffeomorphic?

While the questions above can also be answered in different ways, Heegaard Floer theory provides a unified approach to them, as well as to many other problems.

Heegaard Floer theory was originally defined using pseudo-holomorphic curves (solutions to the nonlinear Cauchy-Riemann equations). More recently, a combinatorial approach to Heegaard Floer theory has been developed, based on grid diagrams. This shows that the resulting invariants are algorithmically computable.

This course is meant as an introduction to low-dimensional topology and Heegaard Floer homology. We will sketch the original definition of the Heegaard Floer invariants (using symplectic geometry), and then focus on the combinatorial aspects.


Knowledge of manifolds and algebraic topology, at the level of Math 225.

References (available online):