Topologia Algébrica

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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 here
    
    

    https://doi.org/10.1007/978-1-4612-4180-5

    if you are on the university network.

    
    

• 14 de fevereiro - 25 de fevereiro
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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.

• 26 de fevereiro - 3 de março
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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.

• 4 de março - 10 de março
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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.

• 11 mars - 17 mars
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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.
    

• 18 mars - 24 mars
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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.
    

• 25 mars - 30 mars

  Non disponible

• 1 avril - 7 avril

  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].

• 8 avril - 14 avril

  Homology - I

  We will continue with homology following Chapter 2 of [Ha].

  
  

• 15 avril - 21 avril

  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.
  

• 22 avril - 28 avril

  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).

  
  
  

• 29 avril - 5 mai

  Homology - III

  We will continue with homology following Chapter 2 of [Ha].