Math 235: Topics in Manifold Theory
Combinatorial Heegaard Floer Homology
Time and Place: MW 1-2:20 pm in MS 7608
Instructor: Ciprian Manolescu
Office: MS 6921
Office Hours: Wednesdays 10-11am and by
Outline: 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
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
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
Prerequisites: Knowledge of manifolds and algebraic topology,
at the level of Math 225.
References (available online):
- P. Ozsvath and Z. Szabo, Holomorphic disks and topological
invariants for closed three-manifolds, Annals of Math. (2)
159 (2004), no. 3, 1027-1158.
- P. Ozsvath and Z. Szabo, Holomorphic disks and three-manifold
invariants: properties and applications, Annals of Math.
(2) 159 (2004), no. 3, 1159-1245.
- P. Ozsvath and Z. Szabo, Holomorphic triangles and invariants
for smooth four-manifolds, Adv. Math. 202 (2006), no. 2, 326-400.
- P. Ozsvath and Z. Szabo, Holomorphic disks and knot
invariants, Adv. Math. 186 (2004), no. 1, 58-116.
- P. Ozsvath and Z. Szabo, Heegaard diagrams and
holomorphic disks, in Different faces of geometry, 301-348, Kluwer/Plenum, New York, 2004.
- P. Ozsvath and Z. Szabo, Knot Floer homology and integer
surgeries, Algebr. Geom. Topol. 8 (2008), no. 1, 101-153.
- S. Sarkar and J. Wang, An
algorithm for computing some Heegaard Floer homologies, Annals of Math.
(2) 171 (2010), no. 2, 1213-1236.
- C. Manolescu, P. Ozsvath and S. Sarkar, A combinatorial description of knot
Floer homology, Annals of Math. (2) 169 (2009), no. 2, 633-660.
- C. Manolescu, P. Ozsvath, Z. Szabo and D. Thurston, On combinatorial link Floer
homology, Geom. Topol. 11 (2007), 2339-2412.
- C. Manolescu and P. Ozsvath, Heegaard Floer homology and integer
surgeries on links, preprint (2010).
- C. Manolescu, P. Ozsvath and D. Thurston, Grid diagrams and Heegaard Floer
invariants, preprint (2009).