Instructor: Ipsita Datta
Email: ipsita DOT datta AT math DOT ethz DOT ch
References: Morse Theory and Floer Homology, by Michèle Audin and Mihai Damian (Springer); Morse Theory, by John Milnor (Princeton University Press)
Listed here are the sections from Audin-Damian we have covered in class. I also list exercises that are relevant to that day's material. Unless mentioned otherwise all section numbers and exercise numbers are from Audin-Damian.| Date | Topics covered | Exercises |
|---|---|---|
| Week 1 | ||
| 20/2 | Introduction and my top reasons for why study Morse homology.
Section 1.1.a Critical Points, Nondegeneracy. Section A.1.a Manifolds Section A.1.e Tangent Vectors, Tangent maps | Exercise 1, 2 |
| 23/2 | Section 1.1.b Examples and Counterexamples
Section 1.2.a Existence of Morse functions Section A.1.b Characterization of Submanifolds | Exercise 3, 4, 5, 6 |
| Week 2 | ||
| 27/2 | Section 1.2.a Existence of Morse Functions
Section 1.2.b Genericness of Morse Functions Section 1.3.a Morse Lemma (excluding proof of Morse lemma) Section 1.3.b Examples of Critical points Examples of degenerate critical points from Milnor - Chapter 2 |
Proof of Lemma 1.2.2 (check details)
Exercise 8,9 |
| 1/3 | Proof of Morse Lemma: Milnor - Lemma 2.1, Lemma 2.2 | Go through the proof of Morse Lemma in Section 1.3.a
Let me know which exercise you want me to discuss on 8/3 |
| Week 3 | ||
| 5/3 | Section 2.1.a Gradients and Pseudo-Gradients
Section 2.1.b Morse charts Section 2.1.c Existence of Pseudo-Gradient fields Section A.4.a Flows |
Lemma 2.4 (and its Proof) from Milnor's book
Exercise 10, 12 |
| 8/3 | Exercise 2
Section 2.1.d Stable and Unstable Submanifolds | |
| Week 4 | ||
| 12/3 | Section 2.1.d Trajectories of the Pseudo-Gradient Field
Section 2.2.a Examples of Stable and Unstable Manifolds Section 2.2.b The Smale Condition |
Exercise 11
Example on page 37 of the other function on the torus, namely, f(x,y) = cos(2\pi x) + cos(2\pi y) Example on page 38 of The Morse Function on P^2(R) |
| 15/3 | Section 2.2.c Existence | Go through details of proof of Theorem 2.2.5 Section 2.2.d |
| Week 5 | ||
| 19/3 | Section 3.1 Definition of the Morse Complex
Section 3.2.a The Space of Broken Trajectories | Example of Morse homology of the Torus as in Section 3.1.c
Examples of Morse homology of Complex Projective Space, Real Projective Spaces as in Section 3.1.c Corollary 3.1.1, Corollary 3.1.2 Exercise 13 |
| 22/3 | Section 3.2.b Compactness | |
| Week 6 | ||
| 26/3 | Section 3.2.c The Structure of a Manifold with Boundary on the Space of Broken Trajectories | |
| 29/3 | Spring break | |
| Week 7 | ||
| 9/4 | Section 3.2.c The Structure of a Manifold with Boundary on the Space of Broken Trajectories
Section 3.3 Orientation, Complex over Z | Exercise 14, 15 |
| 12/4 | Section 3.2.d Manifolds with boundary
Section 3.4 The Homology of the Complex Depends Niether on the Function nor on the Vector Field | |
| Week 8 | ||
| 16/4 | Section 3.4 The Homology of the Complex Depends Niether on the Function nor on the Vector Field | |
| 19/4 | Section 3.5 Cobordisms
Section 4.1 Homology | |
| Week 9 | ||
| 23/4 | Section 4.2 The Künneth Formula Section 4.3 The "Poincaré" Duality | Exercise 16, 17, 19 |
| 26/4 | Section 4.4 Euler Characteristic, Poincaré Polynomial | |
| Week 10 | ||
| 30/4 | Section 4.6 Functoriality of the Morse Homology | |
| 3/5 | Section 4.6 Functoriality of the Morse Homology | |
| Week 11 | ||
| 7/5 | Section 4.6 Functoriality of the Morse Homology
Section 4.7 Long Exact Sequence | |
| 10/5 | Section 2.1.e Topology of the Sublevel Sets: When We Do Not Cross a Critical Value
Section 2.1.f Topology of the Sublevel Sets: When We Cross a Critical Value |
Corollary 2.1.9 (Reeb's theorem) (or Theorem 4.1 from Milnor's book)
Proof of Theorem 2.1.11 (or Theorem 3.1 from Milnor's book) Theorem 3.5 from Milnor's book Homotopy type of CP^n (Page 26 in Milnor's book) |
| Week 12 | ||
| 14/5 | Section 4.5.b Fundamental Group and Homology (Part 1)
Section 4.8.a Fundamental Group and Homology (Part 2) Section 4.9.a Complexes and Cellular Homology | Exercise 20, 21
Optional: Revise or learn Cellular homology from your favourite source. Here are some options: Chapter V (Cellular Decomposition and Cellular Homology) from Dold's Lectures on Algebraic Topology; Section 2.2 (Cellular Homology) from Hatcher's Algebraic Topology; Optional: Related is Simplicial homology. Sources you might like: Chapter V (Cellular Decomposition and Cellular Homology) from Dold's Lectures on Algebraic Topology; Section 2.1 (Simplicial Homology) from Hatcher's Algebraic Topology; Chapter 8 (Simplicial Homology) from Armstrong's Basic Topology. |
| 17/5 | Section 4.9.b Cellular Decomposition Associated with a Morse-Smale Pair | |
| Week 13 | ||
| 21/5 | Section 4.9.c Proof of Theorem 4.9.3 (Isomorphism between cellular homology and Morse homology) | Exercise 24, 25 |
| 24/5 | Section 4.8.c Homology of the Projective Space | Section 4.8.b The Brouwer Fixed Point Theorem Section 4.8.d The Borsuk-Ulam Theorem Exercise 22, 23 |
| Week 14 | ||
| 27/5 | What is Floer homology and the Arnold Conjecture? | Exam preparation exercises |
| 31/5 | Gradient flows as zeroes of a differential operator
References: Section 10.2 The Linearization of the Flow of a Pseudo-Gradient Field, Morse Homology by Matthias Schwarz (Birkhäuser) |