## Math 8062: Schedule and Homework Assignments

The schedule below will be gradually filled out as the class progresses.

Day Topic Pages from Hatcher or other reference Homework
1/18 Overview; Cell complexes P. 1-8
1/20 The fundamental group P. 21-28 Homework 1, due Thursday 1/27
1/25 Fundamental group of the circle P. 29-31
1/27 Brouwer & Borsuk-Ulam theorems P. 31-34
2/1 Induced homomorphismsP. 34-37P. 38-39, #8, 9, 10, 16. Due Tuesday 2/8
2/3 Van Kampen's theorem P. 40-46
2/8 Applications of van Kampen P. 46-52; extra page
2/10Fundamental groups of manifolds Conway's ZIP Proof, Mapping class group Homework 3, due Thursday 2/17
2/15 Intro to covering spaces P. 56-61
2/17 The universal cover P. 63-65 Homework 4, due Thursday 2/24
2/22 Group actions P. 70-74
2/24 Classification of covering spaces P. 61-62, 66-68. Homework 5, due Tuesday 3/15.
3/1 Intro to homology P. 97-101.
3/3 Simplicial homology P. 102-107.
3/15 Singular homology: definitions P. 108-110. P. 131-132 #4, 5, 8, 10, 11. Due Thursday 3/24
3/17 Properties of singular homology P. 110-113.
3/22 Exact sequences, H1 vs. π1 P. 113-114, 166-168.
3/24 Long exact sequence of a pair P. 114-117. P. 123 #14, 16, 17. P. 53 #9 (easier using homology!) Due 3/31
3/29 Excision (in a nutshell) P. 117-124
3/31 Simplicial = singular homology P. 125-130
4/5 Orientations, π1, and homology Lee, p. 329-334 Homework 8, due 4/14
4/7 Degrees of maps P. 134-135
4/12Degrees from smooth theory P. 136, Wikipedia
4/14 Mayer-Vietoris sequence P. 149-153 Homework 9, due 4/21
4/19 Intro to cohomology P. 186-189.
4/21 De Rham cohomology Lee, p. 388-399 (skipping a lot).
4/26 The de Rham theorem Lee, p. 425-430. Homework 10, due 5/5
4/28 Poincaré duality P. 241-248; Lee, p. 432; Wikipedia

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