M8130 Algebraic topology, tutorial 06, 2020 16. 11. 2020
Exercise 1. Use Z/2 coefficients to show, that every cts map f: Sn —> Sn satisfying f(—x) = —f(x) has an odd degree.
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M8130 Algebraic topology, tutorial 06, 2020 16. 11. 2020
Exercise 2. Let <p E Ck(X;R),tp e Cl(^;R). Prove 5{<p U j)) = 5<p U j) + (-l)V U bj). Use r = [e0,..., ek+l+1] G Ck+i+1(X).
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M8130 Algebraic topology, tutorial 06, 2020_16. 11. 2020
Exercise 3. Compute the structure of graded algebra H*(Sn x Sn; Z) for n even and n odd. Use the following:
If Hn{Y\R) is free finitely generated group for all n and (X,A),Y are CW-complexes, then
x: H*(X,A;R)®H*(Y;R) -+ H*(X xY,AxY;R) is an isomorphism of graded rings.
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M8130 Algebraic topology, tutorial 06, 2020
16. 11. 2020
Exercise 4. Prove that there is no multiplication on even dimensional spheres. Multiplication on the sphere Sn is a map m: S" x Sn —> Sn such that there is an element I e Sn satisfying m(x, 1) = x, m(l, x) = x.
Hint: compute m*: H*(Sn) ->• H*(Sn x Sn), describe both rings.
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M8130 Algebraic topology, tutorial 06, 2020
16. 11. 2020
Exercise 5. Show commutativity of the diagram below. Use five lemma to prove that taking any two fi's iso's, the third // is iso as well.
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