Lecture: Seiberg-Witten Floer homology

2 hours lecture + 1 hours exercise session

(In der Prüfungsordnung erscheint diese Vorlesung unter Modul M13: Ausgewählte Themen der Differentialgeometrie)

Dingyu Yang

Winter term 2020/2021

Lecture: Fridays 9:00-11:00 on zoom
Veranstaltungsnummer: 3314430
First lecture: 06.11.2020

Exercise session: Fridays 11:00-13:00 on zoom (every other week)
Veranstaltungsnummer: 33144301
First exercise session: 13.11.2020


Content: Seiberg-Witten Floer homology, also known as Monopole Floer homology, has been definitively treated in Kronheimer-Mrowka's bible called Monopoles and three-manifolds, and is the subject matter of this lecture series.

Seiberg and Witten wrote down a pair of equations naturally defined on oriented closed smooth 4-manifolds which has remarkable properties on the solution spaces, namely compactness and having abelian gauge group. This historically allowed mathematicians to prove some long standing conjectures, such as proof of Thom conjecture by Kronheimer-Mrowka. The idea is to start from a geometric object one is interested in, an oriented closed smooth 4-manifold, one derives from it a moduli/solution space of some natural geometric equations (e.g. Seiberg-Witten equation) with minimal choices, one then applies topological tools to the solution space to get powerful invariants (e.g. Seiberg-Witten invariant) to the original geometric object, which are otherwise not visible from traditional methods in differential geometry or algebraic topology.

One can also look at non-compact complete 4-manifold R x Y where R is real line and Y is an oriented closed 3-manifold, then the Seiberg-Witten equation on it becomes a gradient flow equation of a (Chern-Simons-Dirac) functional, and one can do Morse theory of this functional to get a homology group on Y. Here one studies not the Morse theory on Y, but the Morse theory on a (configuration) space with singularity associated from Y in the same vein as in the last paragraph. The singularity is mild and one uses a version of Morse theory for manifolds with vertical boundary under circle action.

The goal of the course is to introduce the above ideas and ingredients of this beautiful subject systematically and from scratch (relatively speaking). Some familiarity with connection and homology will be helpful and should be studied at least together with this course, but not absolutely essential. Towards the end of the course, we hope to cover an important non-vanishing result for hat (or pronouced as "from") version of homology in chapter 35 of Kronheimer-Mrowka's book, which is partly responsible for lots of ground breaking results in symplectic geometry and topology (since 2007 but as recent as this year) for example via another incarnation called Embedded Contact homology constructed by Hutchings.

Besides covering some of the above, the course hopefully offers a quick peek into interesting research area in gauge theory and moduli spaces. A word or two on some connections, contexts and analogies which are beyond the scope of this course. Seiberg-Witten Floer homology is also manifestation of another computable invariants called Heegaard Floer homology which has been successful in low dimensional and knot theory, and lots of arguments and results there formally apply here as well. It also rekindles study of instanton Floer homology (developed by Floer and Donalson) which precedes and is largely superseded by Seiberg-Witten Floer homology until recent years (e.g. in Kronheimer-Mrowka's work of Khovanov homology detecting the unknot 10 years ago). Enhanced version of Seiberg-Witten theory also disproves triangulation conjecture; and other gauge theoretic invariants in these circle of ideas are under rapid current development in various different fields, for example, Atiyah-Floer conjecture; simple type conjecture; and Donaldson-Thomas type invariants and shifted symplectic structure on moduli spaces etc.

Prerequisites and Exam: I will provide some background motivations and necessary definitions gradually, with useful results carefully stated and packaged into independent units. Prerequisites should include Differential Geometry I+II and Topology I+II, but will otherwise be consciously kept to a minimum, aside from curiosity and a certain willingness to read on the book in the reference. The knowledge assumed in each lecture will be summarized at the beginning of that lecture. Hopefully, the course should appeal to an audience with a variety of different interests and tastes in geometry, topology, physics etc.

The exam format will be flexible and it might consist of a presentation explaining or following up some topic/result in or relevant to this course in a friendly setting with some discussion. Please feel free to email me or talk to me about details. See you on zoom.

Exercise: Mostly compiled from lecture notes. Enough definitions and notations are included so that one does not need to refer to lecture notes, thus do not be alarmed by its length. Optional for understanding.
Exercise (pdf)

Table of Contents: (updated after the lectures, lecture notes are expository mostly borrowing from places freely with credits (borrowing from Banksy: everything here is recycled including ideas), typed version may appear optimistically with a lag of one week or two.)

Lecture 1 (06/11/2020): Notes (pdf)
Lecture 2 (13/11/2020): Notes (pdf)
Lecture 3 (20/11/2020): Notes (pdf)
Lecture 4 (27/11/2020): Notes (pdf)
Lecture 5 (04/12/2020): Notes (pdf)
Lecture 6 (11/12/2020): Notes (pdf)
Lecture 7 (18/12/2020): Notes (pdf)
Merry Xmas everyone~!
Lecture 8 (08/01/2021): Notes (pdf)
Lecture 9 (15/01/2021): Notes (pdf)
Lecture 10 (22/01/2021): Notes (pdf)
Lecture 11 (29/01/2021): Notes (pdf)
Lecture 12 (05/02/2021): Notes (pdf) We have finished the construction today!
Lecture 13 (12/02/2021): Notes (pdf) Non-vanishing result in synopsis
+ sketch of proof for Weinstein conjecture.
Lecture 14 (19/02/2021): Notes (pdf) Volume-detecting
Lecture 15 (26/02/2021): Notes (pdf) smooth closing lemma and non-simplicity theorem.

For complete lecture notes in a single file (updated on 02/03/2021, namely March 2, 2021): Notes (pdf)

Literature: Kronheimer and Mrowka's Monopoles and three-manifolds, New Mathematical Monographs 10, CUP

This part needs more updates and some references are given in the content of lectures and will be updated here.