M8130 Algebraic topology, tutorial 04, 2024
17. 10. 2024
Exercise 1. Prove 5-lemma.
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M8130 Algebraic topology, tutorial 04, 2024_ 17. 10. 2024
Exercise 2. There is a long exact sequence of the triple (X,A,B), i.e. (B C A C X):
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M8130 Algebraic topology, tutorial 04, 2024_ 17. 10. 2024
Exercise 3. Apply previous exercise to the triple (Dk,Sk~1,*), where * is a point.
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M81S0 Algebraic topology, tutorial 04, 2024
17. 10. 2024
Exercise 4. Show that the chain in Ck(Ak,dAk) given by id: Afc Ak is the representative of the generator of
Hk(Ak,dAk)^Z. Q (Use induction and the long exact sequence for triple.)
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M8130 Algebraic topology, tutorial 04, 2024_ 17. 10. 2024
Exercise 5. Using the Mayor-Vietoris exact sequence compute the homology groups of the torus, (note: Vietoris died in 2002, aged 110, remarkable)
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