# Morse Theory and Applications (Spring 2022, Cornell)

Organizers:

## Schedule

We meet on Thursdays from 1-2pm ET.

Online meeting links will be distributed via email.

Topic Speaker Date Location
Morse function definitions (Cpt. 1.1, 1.2), slides Ben 2/3 Zoom
Passing a critical point (Cpt. 1.3) Nicki 2/10 Zoom
No meeting 2/17
Existence of Morse functions (Cpt. 1.4) Nikhil 2/24 Zoom
Handle decompositions (Cpt. 1.3) Hazel 3/3 Zoom
(Un)stable Manifolds (Cpt. 2.1, 2.2) Nikhil 3/10 Malott 206
No meeting 3/17
Manipulating critical points (Cpt. 2.3, 2.5) Hazel and Nikhil 3/24 Malott 206
Heegaard Splittings of 3-Manifolds (Cpt. 2.4) Nikhil 3/31 Malott 206
No meeting 4/7
Consequences of h-cobordism (Cpt. 5.5, 5.6), slides Nikhil 4/14 Malott 206
Morse homology definitions (Cpt. 3.1, 3.2, 3.4) Nicki 4/21 Malott 206
Further Morse homology (Cpt. 3.1, 3.2, 3.4) Nicki 4/28 Malott 206
Morse / de Rham homology isomorphism (Cpt. 3.5) Ben 5/5 Malott 206
Consequences of Morse homology (Cpt. 4) Ben and Nikhil 5/12 Malott 206

## Textbook

We will be using Gille's Castel's Morse Theory, a beautifully illustrated master's thesis following a similar trajectory to Milnor's Lectures on the h-Cobordism Theorem.

## Website

The code for the website is hosted on github. The easiest way to edit the website is to ask the organizers to add you to the github repo. Then clone/update the repo, edit it, and publish your changes with git.

If you're new to git, Udacity has a free git course which teaches the basics in around six simple lessons. The instructor is also very attractive engaging.

## Topics

Any of these items can be changed (or expanded to multiple days) as needed.

1. Cobordisms and Morse functions — Ben and Nikhil
1. Introduce definitions of Morse functions, Hessian and index
2. Prove the Morse Lemma
3. Prove existence/abundance of Morse functions
2. Handle decompositions — Nicki and Hazel
1. Morse functions with zero critical points
2. Morse functions with one critical point
3. Handle attachment and handle decomposition
3. Manipulating critical points — Hazel and Nikhil
1. Gradient-like vector fields and (un)stable manifolds
2. Re-ordering of critical points
3. Cancellation of critical points
4. Translate into the language of handles (handle cancellation and sliding)
4. A day in low dimensions: Heegaard splittings and Lickorish-Wallace — Tappu
1. Heegaard splittings and genus
2. Morse-theoretic proof of Lickorish-Wallace (prove or quote Rokhlin's theorem)
5. A day in high dimensions: Consequences of the h-cobordism theorem — Nikhil
1. Sketch a plan for proving the h-cobordism theorem
2. Prove the classification of higher-dimensional balls
3. Prove the Reeb lemma and the generalized Poincaré conjecture
4. The group of twisted spheres in terms of diffeomorphism groups
5. The higher-dimensional Schoenflies theorem
6. A brief discussion of 4-dimensional problems
6. Morse homology — Nicki and Ben
1. Definition of the chain complex
2. Isomorphism with singular homology
3. Consequences (e.g. Poincaré-Lefschetz duality)
7. Whitney trick
1. Prove the Whitney trick
2. If there is time, mention the strong Whitney embedding theorem?
8. Proof of h-cobordism theorem
1. Cancelling critical points in extreme dimensions
2. Cancelling critical points in middle dimensions
9. Potential further topics
1. Morse (co)homology (Nikhil)
2. Discrete Morse Theory (Alex)
4. Extensions of and follow-ups on the h-cobordism theorem:
2. Successes and failures in 4 dimensions
3. Cerf theory & pseudo-isotopy theorem
4. Kervaire invariant & Milnor-Kervaire description of exotic spheres
5. Cerf theory can also lead up to the Smale conjecture or Kirby calculus
6. Equivariant Morse-Bott theory? (Applications to Riemannian and symplectic)
7. Infinite dimensions: finding geodesics, proving Bott Periodicity, Floer theory
8. Picard-Lefschetz theory? Weinstein manifolds?

### Books

Morse Theory (Milnor) — Part I is a classic introduction to Morse theory. However, the rest of the book is written very specifically to build up to a proof of Bott periodicity, which may not be what we want to cover in this course.

Lectures on the h-Cobordism Theorem (Milnor) — This book obviously has the specific purpose of proving the h-cobordism theorem. But the first five chapters actually give a broader picture of general Morse theory techniques than Milnor's Morse Theory. After that, the topics get more specific, but the coverage of the Whitney trick in Chapter 6 and Morse homology in Chapter 7 are still quite interesting. And the pay-off in Chapter 9 is pretty sweet.

Morse Homology (Banyaga & Hurtubise) — The organizers haven't read this one, but it has been highly recommended. Chapters 8-9 may be less interesting to the non-symplectic folks, but Chapters 2-7 may be of more broad appeal. This book is also a more friendly introduction than Schwarz's Morse Homology, which is supposed to be quite technical and relies more heavily on analysis background.

Differential Manifolds (Kosinski) — The organizers haven't read this one either. This book seems like a mere introduction to differential topology, but the care given to details in the early chapters is useful, while the later chapters prove some really cool results. In fact, Kosinski proves the h-cobordism theorem by Chapter VII and then keeps on going to conclude the book with the Milnor-Kervaire classification of exotic spheres!

Morse Theory and Floer Homology (Audin & Damian) — Floer homology is a flourishing "spin-off" of Morse homology, with powerful applications to low-dimensional topology and symplectic geometry. However, this book is geared specifically towards the symplectic foundations and may be uninteresting to non-symplectic folks.

An Invitation to Morse Theory (Nicolaescu) — The organizers don't know much about it, but some people seem to really like this book. The topics are quite broad and interesting, although some of Chapters 3-5 seem geared towards symplectic and algebraic geometers.

Morse Theory: Smooth and Discrete (Knudson) — This is a very readable introduction to Morse theory. The theory for the smooth part is standard, with many examples. The discrete part is a nice summary of the relatively new theory introduced by Forman, which has found applications in Topological Data Analysis, such as speeding up homological computation.

### Theses

Smooth and Discrete Morse Theory — An Australian colleague wrote his honours thesis on smooth and discrete Morse theory.

Morse Theory and the h-cobordism theorem — Nikhil's honours thesis! On his blog.

### Papers

Cerf Theory — In any of the above books that cover h-cobordism, we learn how to modify Morse functions by adding/cancelling critical points via birth-death singularities. Cerf theory goes in the opposite direction to show that any two Morse functions on a manifold can be related by a sequence of modifications through birth-death singularities. This is fundamental in proving that Kirby calculus and its relatives completely characterize 3- and 4-manifolds. This topic also generalizes to parametrized Morse theory, which studies the space of all Morse functions on a manifold, with applications to the diffeomorphism groups of spheres and the algebraic K-theory of spaces.

• As far as the organizers know, Cerf's original papers have not been translated from French, but there is a nice survey in Hatcher-Wagoner. There is a much briefer survey from Gay, Wehrheim and Woodward.
• Igusa surveys parametrized Morse theory in the short paper "Parametrized Morse Theory and Its Applications." The organizers don't (yet!) have the background needed to understand it, but it looks nice.
• There are also some textbooks that would make for good reading in this direction. Stable Mappings and Their Singularities (Golubitsky & Guillemin) is supposed to be the book to learn about the Thom-Boardman stratification that underlies Cerf theory. To learn some Kirby calculus, 4-Manifolds and Kirby Calculus (Gompf & Stipsicz) seems hard to beat (just look at these reviews).
• Cerf theory and parametrized Morse theory also factor into Hatcher's proof of the Smale conjecture (the inclusion $$O(4) \hookrightarrow \text{Diffeo} (S^3)$$ is a homotopy equivalence).

Morse Homotopy and Field Theory — The organizers don't fully understand what this is about, but we'd love to learn more. Starting with the work of Witten and Kontsevich (hard!), some of their ideas were taken up in really interesting ways by Fukaya, Betz, Cohen, Jones and Segal. This includes getting more information out of Morse homology (e.g. cup product) by studying families of Morse functions simultaneously, as well as graphs composed of flow lines between critical points!

• Fukaya has many early papers on the subject, some more accessible than others.
• There's also Betz-Cohen, Cohen-Jones-Segal and Betz's PhD thesis.
• This paper by Forman describes some related stuff in discrete Morse theory.
• Tadayuki recently used Morse homotopy to disprove the Smale conjecture in four dimensions (the inclusion $$O(5) \hookrightarrow \text{Diffeo}(S^4)$$ is not a homotopy equivalence).

Other topics

• Klein's Coordinate-Free Morse Theory seems like a really neat result.
• Everyone recommends Morse Theory Indomitable, but the organizers have not (yet) been able to follow any part after the entrance of Witten.
• Finite-volume Flows and Morse Theory. David Nadler recommended this as an interesting framework in which to view Morse homology. It indeed seems really cool, but it requires some background on currents.
• PL Morse theory gives a review of Morse theory, PL Morse theory and applications.
• The handlebody and surgery tools that are developed in Morse theory make a wide vista of results more readily accessible. This includes the famed Lickorish-Wallace theorem and some fun papers by our colleague Max Lipton.