Graduate course on Morse theory
University of Warsaw
- Lectures are held on zoom on Fridays 8:30am -- 10:00am CET.
- Zoom link Meeting ID:880 9296 9142. Passcode: the name
of the author of books "Lectures on the h-cobordism theorem" and "Morse theory", which are the probably the best introductory books to Morse theory.
- There's no need of a special access to the course.
- Lectures will be recorded. The recordings will be uploaded to the youtube and the link to youtube will be given on this webpage.
- Lecture notes will be available on Google Drive.
- Classes are on zoom Fridays, 10:15am -- 11:45 am CET. (Zoom schedules for 10am, we start after 15 minutes)
- Zoom link Meeting ID: 865 8884 5260. Passcode: as above
- Classes are not recorded.
- If you're in another time zone, and you want to participate in the classes, it is possible to reschedule the classes or split a class into smaller and shorter groups.
- There will be only an oral online exam in late January/early February.
- During the exam, you need to be able to present solutions to as many problems on the list, as possible. The list of problems is in the section `Classes'
and will be updated during the curse.
- If you're not a student at Warsaw University, you can still pass an exam. Write me an e-mail before Oct. 31rd with cc to a professor at your university, who can help you with formalities. I myself can't guarantee that if you pass the exam, it will count to your curriculum.
Here's a tentative plan of lectures:
- Introduction. Morse lemma, gradient and gradient-like vector fields. Density of Morse functions.
- Morse functions and handle decomposition. Conley index.
- Morse--Smale condition. Morse homology. Proof that Morse homology is the same as singular homology.
- Optionally: Morse flows as currents, based on Harvey and Lawson's paper.
- Handle rearrangement theorem (with proof). Handle cancellation theorem (with a sketch).
- Whitney trick. Sketch of proof of the h-cobordism theorem.
- Rudiments of Cerf theory. Sketch of proof of Reidemeister theorem and Kirby theorem.
- Morse theory for manifolds with boundary. Spliting handle into half-handles.
- Embedded Morse theory. Examples of failure to rearrangeent and cancellations.
- Rising water principle (with proof).
- Immersed Morse theory. Concordance implies homotopy (with a sketch).
- Optionally: definition of linking number by counting trajectories.
Videos of lectures.
- Lecture 1 Morse lemma, density of Morse functions.
- Lecture 2 Gradient-like vector fields.
- Lecture 3 Crossing critical points.
- Lecture 4 Vector Field Integration Lemma.
- Lecture 5 Vector Field Integration Lemma and Elementary Rearrangement Theorem.
- Lecture 6 Isotopy Injection Lemma and Cancellation Theorem
- Here's a list of problems for classes
- The list will be successively updated
- During classes we will solve problems or discuss the solutions.
- Feel free to ask for solving problems or propose your own solutions.
The key point to understand Morse theory is a deep knowledge of ordinary differential equations. By deep I don't mean any particular statement, but a rather a working
knowledge of basics.
Additionally, some knowledge of topology is helpful. You need to know what is a manifold, what is a tangent bundle, what is homotopy. It helps if you know what is a chain
complex and what is singular homology.
- Milnor's "Lectures on h-cobordism theorem" is a must read book on Morse theory. Note, this book has more about Morse theory then the book "Morse theory" by the same author.
- As a supplementary book I recommend Liviu Nicolaescu's "Invitation to Morse theory".
- Morse homology will be based on Salomon's paper (Morse theory, Conley index and Floer homology) and on Banyaga--Hurtubise book.
- For rudiments of Cerf theory I do not quite recommend Cerf's paper. It is better to get familiar with more modern approach to singularity theory, like the one in the book of Arnold, Varchenko, Gussein-Zade.
- Morse theory for manifolds with boundary is based on the book of Kronheimer and Mrowka (Section 2.4) and on my paper with A. Nemethi and A. Ranicki.
- Embedded Morse theory is based on my paper with Mark Powell.
- Immersed Morse theory and rising water principle will be based on my papers with Mark Powell and Peter Teichner. I hope some of them will be posted on the arxiv soon.