M8130 Algebraic topology, tutorial 03, 2024_10.10.2024
Exercise 1. Show dd = 0. Use formula e^+1 o^ = o e\, where i ^ j. The definition force Cn(X), a: An -> X, is
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M8130 Algebraic topology, tutorial 03, 2024_ 10.10.2024
Exercise 2. Simplicial homology ofdA2;
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M8130 Algebraic topology, tutorial 03, 2024
10.10.2024
Exercise 3. Simplicial complex, model of torus, compute differentials and homology.
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M8130 Algebraic topology, tutorial 03, 2024
10.10.2024
Exercise 4. Prove the first criterion of homotopy equivalence.
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M8130 Algebraic topology, tutorial 03, 2024
10.10.2024
Exercise 5. S2 V S1 ~ S2/S° (using First criterion)
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M81S0 Algebraic topology, tutorial 03, 2024_ 10.10.2021
Exercise 6. Let i: A X is a cofibration, show X/A ~ X U CA = Ci. (using First criterion)
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M8130 Algebraic topology, tutorial 03, 2024
10.10.2024
Exercise 7. Application of the criterion: two types of suspensions, unreduced and reduced. Unreduced suspension: SX = Ix//~, where      0) ~ (x2,0), (xi, 1) ~ (x2, 2). Reduced suspension: SX = SX/{x0} x 7 = (X, x0) A (5\ s0) The criterion says, that if {x0} ^ X is a cofibmtion, then SX ~ SX.
/ ~ {{x0,t),t G 1} C SX —> 5X/{x0!*),t e /} = SX
A/
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M8130 Algebraic topology, tutorial 03, 2024_10.10.2024
Exercise 8. There is a lemma, that says: Given the following diagram, where rows are long exact sequences and m is iso,
tf„ —Ln —M„ —^ -> L„_i -> Mre_!
we gei a font? exact sequence
where dt = hom 1 o g. Show exactness in Ln © Kn and in Ln.
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