Course Outlines

There will be three series of lectures, on The Seiberg-Witten Equations and Symplectic Geometry, Heegaard Floer Theory, and Sutured Manifolds. The Heegaard Floer and sutured manifold lectures will be particularly closely linked, and each series will rely on earlier talks in the other.

Heegaard Floer series:


Heegaard Floer theory is an analogue of Seiberg-Witten theory, defined in terms of holomorphic curves. This lecture series we will focus on the definitions of (some of) the Heegaard Floer invariants, as well as their basic properties. Later lectures in the series will focus on bordered Floer homology, an extension of Heegaard Floer theory, and how it fits in with the rest of the subject.

Sutured manifold series:


A sutured manifold is a 3-manifold with boundary divided into an "upper" and a "lower" piece. They naturally arise in the study of foliations and contact structures. In this companion series to the Heegaard Floer series, we will use sutured manifolds to show how Heegaard Floer homology gives geometric information.

Heegaard Floer 1: "Morse theory and 3-manifolds."
Morse functions, cell complexes. Heegaard decompositions and diagrams for 3-manifolds. Morse homology with a view towards Floer homology.

References:
J. Milnor, Morse Theory.
J. Milnor, Lectures on the h-Cobordism Theorem.
M. Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), http://math.berkeley.edu/~hutching/teach/276-2010/mfp.ps
A. Banyaga and D. Hurtubise, Lectures on Morse homology, http://www.ams.org/mathscinet-getitem?mr=2145196
M. Schwarz, Morse homology. http://www.ams.org/mathscinet-getitem?mr=1239174

Sutured manifolds 1: "Knots and Grid Diagrams."
Grid diagrams and knot Floer homology. Knot genus. Fibered knots. Thurston norm. Fibered homology classes.

References:
W. Thurston, "A norm for the homology of 3-manifolds." http://www.ams.org/mathscinet-getitem?mr=823443
C. Manolescu, P. Ozsvath and S. Sarkar, "A combinatorial description of knot Floer homology." http://www.ams.org/mathscinet-getitem?mr=2480614
C. Manolescu, P. Ozsvath, D. Thurston and Z. Szabo, "On combinatorial link Floer homology." http://www.ams.org/mathscinet-getitem?mr=2372850

Heegaard Floer 2: "Holomorphic disks in Heegaard diagrams."
Formal structure of Heegaard Floer homology. Definition of Heegaard Floer invariants of 3-manifolds, knots in 3-manifolds. Cylindrical formulation, computations.

References:
P. Ozsvath and Z. Szabo, "An introduction to Heegaard Floer homology." (Survey.) http://www.ams.org/mathscinet-getitem?mr=2249247
(Available at: http://www.claymath.org/publications/Floer_Homology/szabo.pdf )
P. Ozsvath and Z. Szabo, "Lectures on Heegaard Floer homology." (Survey.) http://www.ams.org/mathscinet-getitem?mr=2249248
P. Ozsvath and Z. Szabo, "Holomorphic disks and topological invariants for closed three-manifolds." http://www.ams.org/mathscinet-getitem?mr=2113019
R. Lipshitz, "A cylindrical reformulation of Heegaard Floer homology." http://www.ams.org/mathscinet-getitem?mr=2240908

Sutured manifolds 2: "Sutured Manifolds..."
Definition of sutured 3-manifolds. Surface decompositions of sutured 3-manifolds, sutured hierarchies. Taut foliations.

References:
D. Gabai, "Foliations and the topology of 3-manifolds." (Survey.) http://www.ams.org/mathscinet-getitem?mr=682826
D. Gabai, "Foliations and the topology of -manifolds." http://www.ams.org/mathscinet-getitem?mr=723813
M. Scharlemann, "Sutured manifolds and generalized Thurston norms." http://www.ams.org/mathscinet-getitem?mr=992331
M. Scharlemann, "Lectures on the theory of sutured 3-manifolds." http://www.ams.org/mathscinet-getitem?mr=1098719

Heegaard Floer 3: "Bordered Floer basics."
Formal structure of bordered Floer homology. Definition of the algebras, modules. Examples. Planar grid diagrams toy model. Auroux's Fukaya-categorical formulation.

References:
R. Lipshitz, P. Ozsvath and D. Thurston, "A tour of bordered Floer theory." (Survey.) http://arxiv.org/abs/1107.5621
R. Lipshitz, P. Ozsvath and D. Thurston, "Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology." (Survey.) http://www.ams.org/mathscinet-getitem?mr=2500575 R. Lipshitz, P. Ozsvath and D. Thurston, "Bordered Heegaard Floer homology: Invariance and pairing." http://arxiv.org/abs/0810.0687 D. Auroux, "Fukaya categories of symmetric products and bordered Heegaard-Floer homology." http://www.ams.org/mathscinet-getitem?mr=2755992

Sutured manifolds 3: "...and Heegaard Floer theory"
Sutured Floer homology, bordered-sutured Floer homology. Surface decomposition theorem for sutured Floer homology. Heegaard Floer homology detects genus and fiberedness.

References:
A. Juhasz, "Holomorphic discs and sutured manifolds." http://www.ams.org/mathscinet-getitem?mr=2253454
A. Juhasz, "Floer homology and surface decompositions." http://www.ams.org/mathscinet-getitem?mr=2390347
R. Zarev, "Bordered Floer homology for sutured manifolds." http://arxiv.org/abs/0908.1106

Heegaard Floer 4: "The analysis in bordered Floer homology"
Moduli spaces of holomorphic curves. Compactness. Proof of the pairing theorem.

References:
R. Lipshitz, P. Ozsvath and D. Thurston, "Bordered Heegaard Floer homology: Invariance and pairing." http://arxiv.org/abs.0810.0687

Heegaard Floer 5: "Duality properties in (bordered) Heegaard Floer homology."
Orientation reversal. Conjugation on Spin-C structures. D is A. Hochschild homology. Serre functor. Koszul duality.

References:
P. Ozsvath and Z. Szabo, "Holomorphic disks and three-manifold invariants: properties and applications." http://www.ams.org/mathscinet-getitem?mr=2113020
R. Lipshitz, P. Ozsvath and D. Thurston, "Heegaard Floer homology as morphism spaces." http://arxiv.org/abs/1005.1248

Heegaard Floer 6: "Computing with bordered Floer homology."
Sketch of how to compute HF-hat using bordered Floer homology. Computing HF-hat 4-manifold invariants this way.

References:
R. Lipshitz, P. Ozsvath and D. Thurston, "Computing HF^ by factoring mapping classes." http://arxiv.org/abs/1010.2550

The Seiberg-Witten Equations and Symplectic Geometry

These lectures will focus on the Seiberg-Witten monopole equations in 4, 3 and 2 dimensions, emphasizing their connections to symplectic geometry.

One such connection is that, in a symplectic 4-manifold, monopoles localize - in the "Taubes limit" - on holomorphic curves. Another is that monopoles on the product of two Riemann surfaces correspond, in the "adiabatic limit" where one factor is stretched, to holomorphic maps from the stretched surface to a symmetric product of the other. A third is that the boundary values of 3-dimensional monopoles trace out a Lagrangian submanifold of the symplectic vector space of fields on the boundary.

I hope to outline how such ideas lead to an isomorphism of monopole Floer homology with Heegaard Floer homology and to conjectural extensions of the monopole TQFT down to 2 dimensions.

Recommended reading:
P. Kronheimer and T. Mrowka, Monopoles and three-manifolds. Chapter I: Outlines. Cambridge University Press.
S. Donaldson, The Seiberg-Witten equations and 4-manifold topology. Bulletin of the AMS, 33 (1): 45-70.
M. Hutchings, Taubes's proof of the Weinstein conjecture in dimension three. Bulletin of the AMS, 47 (1): 73–125. arXiv: 0906.2444

Outline:

  1. What gauge theory has done for us.
    An overview of the applications of the theories of Donaldson, Seiberg-Witten and Ozsvath-Szabo. What these theories see and what they do not; differences between them. Gauge-theory moduli spaces in action.
  2. The vortex equations and the Seiberg-Witten equations.
    Vortices and symmetric products. The monopole equations in 4-dimensions. The symplectic case: localization on holomorphic curves. Monopoles as holomorphic maps to vortex space.
  3. Monopole invariants in 4 and 3 dimensions.
    Invariants for closed 4-manifolds. Monopole Floer homology for 3-manifolds; relation to 4-manifold invariants. Reducibles and wall-crossing.
  4. Monopoles as Lagrangian intersections.
    Vortex spaces are symplectic. Immersed Lagrangian submanifolds of asymptotic values. Heegaard = monopole Floer as an "Atiyah-Floer" principle.
  5. Down to 2 dimensions.
    Extended TQFT and Fukaya categories of symmetric products. Lagrangian correspondences and quilted Floer homology; Lekili-Perutz extension of Heegaard theory.

Last modified Saturday October 29th, 2011