math 545: manifolds

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math544.zulipchat.com

• Instructor: Kyle Ormsby
• TA: Alex Waugh
• Text: Introduction to Topological Manifolds (2nd ed.) & Introduction to Smooth Manifolds
• Course Information & Syllabus
• Problem session: Tuesdays 13:30-14:30 in the graduate lounge
• Drop-in hours: W 11:30-12:30 with Alex in C-132, Th 15:00-16:00 (and by appointment) with Kyle in C-442

Students are strongly encouraged to typeset their work in $\LaTeX$. You can get started easily with Overleaf. To facilitate your typesetting, for (nearly) all URL’s on this site ending in .pdf, you can change the suffix to .tex to access the source code.

week 1

4 January - 6 January

Wednesday: Welcome & warmup

• Reading: Syllabus
• Handout, Notes, Recording

Friday: Covering maps

• Reading: ITM 277-282
• Notes, Recording
• Homework due: none

week 2

9 January - 13 January

Monday: Lifting & monodromy

• Reading: ITM 283-293
• Notes, Recording

Wednesday: Covering homomorphisms & universal covering space

• Reading: ITM 294-301
• Notes, Recording

Friday: Automorphisms of coverings

• Reading: ITM 307-317
• Notes, no recording (sorry!)
• Homework due: week02

week 3

16 January - 20 January

Monday: Martin Luther King Jr. Day, no class

Wednesday: Classification theorem and examples

• Reading: ITM 318-333
• Notes, Recording

Friday: Homology

• Reading: ITM 339-346
• Notes, Recording
• Homework due: week03 (For typesetting purposes, the figure in the homework is available here.)

week 4

23 January - 27 January

Monday: Homology & $\pi_1$

• Notes, Recording
• Reading: ITM 347-354

Wednesday: Mayer-Vietoris

• Notes, Recording
• Reading: ITM 355-368

Friday: Homology & cohomology

• Reading: ITM 364-378
• Notes, Recording
• Homework due: week04

week 5

30 January - 3 February

Monday: Smooth manifolds

• Reading: ISM 1-29
• Notes, Recording

Wednesday: Smooth functions and smooth maps

• Reading: ISM 32-40
• Notes, Recording

Friday: Partitions of unity

• Reading: ISM 40-47
• Notes, Recording
• Homework due: week05

week 6

6 February - 10 February

Monday: Tangent vectors

• Reading: ISM 50-59
• Notes, Recording

Wednesday: Tangent bundle

• Reading: ISM 60-67
• Notes, Recording
• Midterm exam distributed

Friday: Velocity

• Reading: ISM 68-74
• Notes, Recording
• Midterm exam due; no homework

week 7

13 February - 17 February

Monday: Maps of constant rank

• Reading: ISM 77-84
• Notes, Recording

Wednesday: Embeddings

• Reading: ISM 85-90
• Notes, Recording

Friday: Submersions and smooth covering maps

• Reading: ISM 90-95
• Notes, Recording (audio starts at 8:00)
• Homework due: week07

week 8

20 February - 24 February

Monday: Presidents’ Day, no class

Wednesday: Embedded submanifolds

• Reading: ISM 98-107
• Notes, Recording

Friday: More submanifolds

• Reading: ISM 108-119
• Notes, Recording
• Homework due: week08

week 9

27 February - 3 March

Monday: Sard’s theorem

• Reading: ISM 125-135
• Notes, Recording

Wednesday: Whitney embedding

• Reading: ISM 131-137
• Notes, Recording

Friday: Tubular neighborhoods

• Reading: 137-143
• Notes, Recording
• Homework due: week09

week 10

6 March - 10 March

Monday: Transversality

• Reading: 143-147
• Notes, Recording

Wednesday: Activities / review

• Handout, Notes

Friday: Review

• Notes, Recording
• Homework due: week10

finals week

13 March - 17 March

• Practice problems.
• Sign up. (Slots are 45 minutes but the exam will only take place during the first 30 minutes of your appointment.)