Math 6510: Differentiable manifolds


Instructor: Mladen Bestvina
Office: JWB 210
Office hours: By appointment
Meets: MWF 10:45-11:35 in JWB 333
Midterm: Friday, Oct. 4
Final: Monday, December 9, 2024, 10:30 – 12:30
Text: There are many textbooks on differentiable manifolds. We'll mainly be using

but here are some others that you may find useful:

The main drawback of Guillemin-Pollack is that manifolds are defined as subsets of Euclidean space instead of the more modern and standard definition via charts. To bridge this difference I wrote some notes. The last few weeks of classes we'll switch to another book (probably Spivak) to cover Lie brackets, Lie derivatives, foliations etc.

Homework: It will be assigned weekly. You are encouraged to work in groups, but what you write should be your own work and you should list the other people in your group. It will be due every week on Mondays at 9 am and you should turn it in through canvas in latex. Late homework is not accepted but the lowest two scores are dropped from the count. You should read the assigned reading for the week before the corresponding lecture. I will also typically give out several problems each week that you are not required to turn in.

Problem sessions: We'll have a problem session every other week where you are expected to present solutions to unassigned problems. It'll be on Fridays 3-4 in LCB 222. The first session: 8/30. The signup sheet is here.

Grading: The final grade is based on homework (30%), problem session activity (10%), the midterm (20%) and the final (40%).

Week
 Reading
 Homework
 Due Date
1
 Notes Ch 1, G-P 1.1
 1,4,6,9,12 from hw01, if it's helpful here is the tex file.
 8/26
2
 Notes Ch 2-3, G-P 1.2-1.4  (I periodically update the notes, you may want to download again)
1,4,11,12,13 from hw02 and the latex file.
 9/3
3
 It's a short week. We'll do some odds and ends and start with partitions of unity
 1,3,9 from hw03 and the latex file
 9/9
4
 partitions of unity and transversality, Sard's theorem
 1,3,7,8,9 from hw04 and the latex file
 9/16
5
 Tangent bundles, Whitney embedding, Morse functions
 1,2,3,5,7 from hw05 and the latex file
 9/23
6
 Manifolds with boundary, normal bundles, tubular neighborhood theorem
 4,5,6,7,10 from hw06 and the latex file
 9/30
7
 Intersection theory mod 2.
No homework. Get ready for the exam on Friday.
8
 Oriented intersection theory, Poincare-Hopf
 1,3,6,8,11 from hw07 and the latex file
 10/21
9
 Poincare-Hopf, differential forms and integration. Read section 4.2 on the exterior algebra in Guillemin-Pollack (or equivalent).
 1,2,6c (you are allowed to use 6a,6b), 8, 11 from hw08 and latex file
 10/28
10
 Exterior derivative, Stokes, de Rham cohomology
 1,2,8,9,10 from hw09 and latex file
 11/4
11
 Poincare lemma, degree theorem, computations of de Rham cohomology
 2,3,4,5,10 from hw10 and latex file
 11/11
12
 Thom class, cup product (informally), integral curves
 1,2,4,5 from hw11 and latex file
 11/18
13
 Flows, Lie derivatives
 1,3,4,5,7 from hw12 and latex file
 11/25
14
 Plane fields, integral manifolds, Frobenius theorem
 hw13, but nothing to turn in


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