winter 2023

math 545 · manifolds

Covering spaces, monodromy, the classification of coverings, and a first look at homology.

teaching assistant: Alex Waugh
textbook: Introduction to Topological Manifolds (2nd ed.) & Introduction to Smooth Manifolds
syllabus: Course Information & Syllabus
problem session: Tuesdays 13:30-14:30 in the graduate lounge
office 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 $L A T E X$ . 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.

jump to this week ↓30 meetings

week 14 January - 6 January

wed jan 4

Welcome & warmup
readingSyllabus
notesnotes❦handout
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fri jan 6

Covering maps
readingITM 277-282
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noteHomework due: none

week 29 January - 13 January

mon jan 9

Lifting & monodromy
readingITM 283-293
notesnotes
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wed jan 11

Covering homomorphisms & universal covering space
readingITM 294-301
notesnotes
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fri jan 13

Automorphisms of coverings
readingITM 307-317
notesnotes
homework dueproblem set

week 316 January - 20 January

mon jan 16no class

Martin Luther King Jr. Day, no class

wed jan 18

Classification theorem and examples
readingITM 318-333
notesnotes
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fri jan 20

Homology
readingITM 339-346
notesnotes
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homework dueproblem set

week 423 January - 27 January

mon jan 23

Homology & $π_{1}$
readingITM 347-354
notesnotes
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wed jan 25

Mayer-Vietoris
readingITM 355-368
notesnotes
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fri jan 27

Homology & cohomology
readingITM 364-378
notesnotes
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homework dueproblem set

week 530 January - 3 February

mon jan 30

Smooth manifolds
readingISM 1-29
notesnotes
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wed feb 1

Smooth functions and smooth maps
readingISM 32-40
notesnotes
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fri feb 3

Partitions of unity
readingISM 40-47
notesnotes
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homework dueproblem set

week 66 February - 10 February

mon feb 6

Tangent vectors
readingISM 50-59
notesnotes
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wed feb 8

Tangent bundle
readingISM 60-67
notesnotes
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noteMidterm exam distributed

fri feb 10

Velocity
readingISM 68-74
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noteMidterm exam due; no homework

week 713 February - 17 February

mon feb 13

Maps of constant rank
readingISM 77-84
notesnotes
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wed feb 15

Embeddings
readingISM 85-90
notesnotes
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fri feb 17

Submersions and smooth covering maps
readingISM 90-95
notesnotes
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homework dueproblem set

week 820 February - 24 February

mon feb 20no class

Presidents' Day, no class

wed feb 22

Embedded submanifolds
readingISM 98-107
notesnotes
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fri feb 24

More submanifolds
readingISM 108-119
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homework dueproblem set

week 927 February - 3 March

mon feb 27

Sard's theorem
readingISM 125-135
notesnotes
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wed mar 1

Whitney embedding
readingISM 131-137
notesnotes
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fri mar 3

Tubular neighborhoods
reading137-143
notesnotes
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homework dueproblem set

week 106 March - 10 March

mon mar 6

Transversality
reading143-147
notesnotes
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wed mar 8

Activities / review
notesnotes❦handout

fri mar 10review

Review
notesnotes
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homework dueproblem set

finals week13 March - 17 March

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