Topologia Algébrica
Agenda semanal
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Textbook
We shall mainly follow Hatcher's book, available here:https://pi.math.cornell.edu/~hatcher/AT/ATpage.html
cited as [Ha] below. We shall also use Fulton's book, cited as [Fu] below, which can be downloaded for free herehttps://doi.org/10.1007/978-1-4612-4180-5
if you are on the university network.
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Cell complexes; fundamental group and covering spaces - 1
I covered (approximately) the two first sections of Chapter 0 of [Ha] (homotopy and cell complexes) and Section 1.1.
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Fundamental group and covering spaces - 2
I plan to cover Section 1.2 of [Ha]: the Seifert-Van Kampen Theorem and applications.
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Fundamental group and covering spaces - 3
I will finish Sec. 1.2 of [Ha] and briefly explain the classification of compact surfaces (i.e., two-dimensional topological manifolds). I will then continue with Sec. 1.3 of Hatcher on covering spaces.
On Thursday (7 March) we will discuss the following exercises from [Ha]: Sec. 1.1: 5, 6, 11, 18; Sec. 1.2: 4, 6, 11.
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Fundamental group and covering spaces - 4
We discussed the exercises and continued the treatment of covering spaces following Section 1.3 of [Ha], concluding with the construction of the universal covering and the proof that it is a G-covering for the group of deck transformations.
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Fundamental group and covering spaces - 5
I will finish Section 1.3 of [Ha] and prove the Seifert-Van Kampen theorem following Chapter 14 of [Fu].
Suggested exercises from Section 1.3 of [Ha]: 8, 9, 10, 15, 18.
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Exercises and homology - 0
On Thursday we will discuss any questions arising from the exercises from Section 1.3 of [Ha] suggested above. If time permits, I will then continue with homology following Chapter 2 of [Ha].
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Homology - I
We will continue with homology following Chapter 2 of [Ha].
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No classes
I will be away at a conference, so there are no classes this week. The classes will be replaced later.
Suggested exercises from Hatcher, Sec. 2.1: 1, 4, 5, 7. I also suggest calculating the simplicial homology of RP^2 using the Delta-complex structure from the text. -
Homology - II
We will continue with homology following Chapter 2 of [Ha]. Tuesday's class will be from 2pm-5pm. Note that there is no class on 25 April (national holiday).