Math 276: Seiberg-Witten Floer theory

Instructor

Textbooks

Plan for the course

• Introduction/overview.
• Gentle review of background: Morse homology, connections on principal bundles, spin-c structures and Dirac operators, etc.
• Seiberg-Witten invariants of 4-manifolds (in brief).
• Seiberg-Witten Floer theory of 3-manifolds (we will try to understand as much as we can, but certainly not all, of the book by Kronheimer-Mrowka).
• Applications (as time permits), optional student presentations.

Lecture summaries and references

• (8/31) Introduction to where Seiberg-Witten Floer theory comes from, what it is good for, and how it relates to other things. Here are references for some of the topics mentioned (reading these is completely optional):
  • Donaldson and Kronheimer, The geometry of four-manifolds. A beautiful book about Donaldson invariants of four-manifolds (the predecessor of Seiberg-Witten invariants). The first chapter gives a nice overview of four-manifold theory and how it was revolutionized by Donaldson theory. The beginning of the second chapter has a nice concise treatment of connections on principal bundles (which we will be discussing shortly).
  • Donaldson, Floer homology groups in Yang-Mills theory: all about instanton Floer homology of three-manifolds (used recently by Kronheimer-Mrowka to prove that Khovanov homology detects the unknot).
  • Witten, Monopoles and four-manifolds. The bombshell which introduced Seiberg-Witten invariants of four-manifolds.
  • Kronheimer-Mrowka, The genus of embedded surfaces in the projective plane, available here. Quick proof of the Thom conjecture using Seiberg-Witten theory.
  • Not sure where best to start with Heegaard Floer homology, but Ozsvath-Szabo have written some survey articles.
  • Here is a survey article on Taubes's proof of the Weinstein conjecture using Seiberg-Witten theory.
  • Finally, you might start looking at Part I (Outlines) of Kronheimer-Mrowka's book. The next few lectures should help with this.
• (9/2) Introduced Morse homology.
  • For an outline see Kronheimer-Mrowka sections 2.1-2.2 (whose conventions I tried to follow).
  • There are brief discussions of Morse homology in my Weinstein conjecture survey section 5, and in section 2 of this paper.
  • For a longer introduction you can see these notes (which desperately need revising, sorry).
  • For details of the analysis see Matthias Schwarz, Morse homology.
• (9/7) More about Morse homology: continuation maps, Poincare duality, local coefficients.
• (9/9) Brief review of obstruction theory and Chern classes. Started reviewing connections and curvature.
  • For very basic obstruction theory, these notes might be helpful.
  • For Chern classes via curvature of connections, see Appendix C of Milnor and Stasheff, Characteristic classes. (Beware that they have some strange sign conventions with curvature, I think because their connection matrix is the transpose of what it should be.)
• (9/14) More about connections, curvature, and Chern classes. Principal bundles.
• (9/16) More about connections on principal bundles. Started discussing spin structures.
  • A beautiful reference on spin structures is the book Spin geometry by Lawson and Michelson. (Note that there is a mistake on the top of page 392: it should say that spin-c structures on X, when they exist, are an affine space over H^2(X;Z).)
• (9/21) Clifford algebras, spinor bundles.
• (9/23) Dirac operators.
• (9/28) Basics of elliptic operators. See e.g. Lawson-Michelson, part III (you might want to skip over the hard parts on a first reading).
• (9/30) Bochner-Lichnerowicz-Weitzenbock formula. Spin-c structures.
• (10/5) Finally wrote down the (4d) Seiberg-Witten equations. See Kronheimer-Mrowka section 1.3.
• (10/7) Started compactness by proving the a priori bound on Seiberg-Witten solutions and reviewing Sobolev spaces.
• (10/12) Completed the proof that the moduli space of Seiberg-Witten solutions (modulo gauge transformations) on a closed 4-manifold is compact. (If I recall correctly, the full details of this are in John Morgan's book on Seiberg-Witten theory.)
• (10/14) Smoothness of the moduli space of solutions to the perturbed equations for generic perturbations.
• (10/19) Dimension and orientation of the moduli space. Definition of the 4d Seiberg-Witten invariants. Proof of the finiteness property.
• (10/21) Additional basic properties of Seiberg-Witten invariants of 4-manifolds.
• (10/26) Started discussing Taubes's theorem on Seiberg-Witten invariants of symplectic four-manifolds, following these notes on Taubes's 1997 lectures.
• (10/28) Finished Taubes's theorem.
• (11/2) Started the proof of the adjunction inequality and introduced three-dimensional Seiberg-Witten theory. See Kronheimer-Mrowka, chapter 4.
• (11/4) Neck stretching, energy bounds. For the general compactness theorem see Kronheimer-Mrowka, chapter 5.
• (11/9) Class cancelled. To make up for this, we will have (optional) meetings at the usual time and place on 12/7 and 12/9.
• (11/11) Holiday.
• (11/16) Completed discussion of the adjunction inequality. Introduction to Seiberg-Witten Floer homology. See Kronheimer-Mrowka chapter 3.
• (11/18) The ambiguity in the functional, the ambiguity in the index, and how these difficulties cancel each other out. For the business about index and spectral flow, see Kronheimer-Mrowka section 14.2. See also Robbin and Salamon, "The spectral flow and the Maslov index".
• (11/23) Morse theory for manifolds with boundary, as needed to define Seiberg-Witten Floer homology in the presence of reducibles. See Kronheimer-Mrowka chapters 2 and 22.
• (11/25) Holiday.
• (11/30) More about Seiberg-Witten Floer homology, mostly from Kronheimer-Mrowka chapter 22.
• (12/2) Introduction to Taubes's proof of the Weinstein conjecture. There is more about this in my survey article.
• (12/7) Presentation by Georg Oberdieck.
• (12/9) Presentations by Yi Liu and Amy Spivak.