fall 2022

math 544 · topology

Topological spaces, manifolds, connectedness and compactness, and the fundamental group.

teaching assistant: Alex Waugh
textbook: Introduction to Topological Manifolds (2nd ed.) by Lee
syllabus: Course Information & Syllabus
problem session: Tuesdays 13:00-14:00 in the graduate lounge
office hours: W 12:30--13:30 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 ↓33 meetings

week 128 September - 30 September

wed sep 28

Welcome, warmup, and course policies
readingSyllabus (finalized Friday)
notesnotes❦handout

fri sep 30

Topological spaces
readingpp. 19–37
supplementalpp. 1-17
notesnotes
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homework dueproblem set

week 23 October - 7 October

mon oct 3

Manifolds
readingpp. 38–45
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wed oct 5

Subspaces, products
readingpp. 49–64
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fri oct 7

Quotients
readingpp. 65–71
notesnotes
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homework dueproblem set

week 310 October - 14 October

mon oct 10

More quotients, gluing, group actions
readingpp. 71–81
notesnotes
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wed oct 12

Connectedness
readingpp. 85–93
notesnotes
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fri oct 14

Compactness
readingpp. 94–103
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homework dueproblem set

week 417 October - 21 October

mon oct 17

Local and paracompactness
readingpp. 104–114
notesnotes
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wed oct 19

Partitions of unity and proper maps
readingpp. 114–121
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fri oct 21

Cell and CW complexes
readingpp. 127–142
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homework dueproblem set

week 524 October - 28 October

mon oct 24

One-dimensional manifolds
readingpp. 143–146
notesnotes
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wed oct 26

Surfaces via polygons
readingpp. 159–172
notesnotes
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fri oct 28

Surface classification
readingpp. 173–177
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homework dueproblem set

week 631 October - 4 November

mon oct 31

Euler characteristic & orientability
readingpp. 178–180
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wed nov 2

Homotopy & the fundamental group $π_{1}$
readingpp. 183–192
notesnotes
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noteTake-home midterm distributed at 9am on Gradescope

fri nov 4

More on $π_{1}$
readingpp. 192–200
notesnotes
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noteTake-home midterm due by 5pm

week 77 November - 11 November

mon nov 7

Homotopy equivalence
readingpp. 200–207
notesnotes
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wed nov 9

Higher homotopy groups, categories & functors
readingpp. 208–213
notesnotes
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fri nov 11no class

Veterans Day, no class
homework dueproblem set

week 814 November - 18 November

mon nov 14

Lifting properties of the circle
readingpp. 217–223
notesnotes
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wed nov 16

$π_{1} (S^{1}) ≅ Z$
readingpp. 224–229
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fri nov 18

Free products, free groups, & group presentations
readingpp. 233–247
notesnotes
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homework dueproblem set

week 921 November - 25 November

mon nov 21

Seifert-van Kampen Theorem: statement & applications
readingpp. 251–263
notesnotes
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wed nov 23no class

Thanksgiving holiday, no class

fri nov 25no class

Thanksgiving holiday, no class

week 1028 November - 2 December

mon nov 28

$π_{1}$ of compact surfaces & classification
readingpp. 264–267
notesnotes
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wed nov 30

Proof of Seifert-van Kampen
readingpp. 268–272
notesnotes
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fri dec 2

Knot groups
homework dueproblem set

week 115 December - 9 December

mon dec 5review

Review & moduli of pentagons
notesnotes
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wed dec 7review

Review & exponential topologies
notesnotes
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fri dec 9review

Review, configuration spaces, & braids
notesnotes
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homework dueproblem set

finals week12 December - 16 December

noteThirty-minute oral exams. Scheduling link available on Zulip. - Practice problems