M8130 Algebraic topology, tutorial 11, 2020 6. 1. 2020
Exercise 1. If (X,A) is relative CW-complex such that there are no cells in dimension < n in X \A, then (X, A) is n-connected.
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M8130 Algebraic topology, tutorial 11, 2020
6. 1. 2020
Exercise 2. Let [X,Y] denote a set of homotopy classes of maps from X to Y. If (X,x0) is a CW-complex and Y is simply connected, then [X,Y] = [(X,x0), (Y,y0)].
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M8130 Algebraic topology, tutorial 11, 2020 6. 1. 2020
Exercise 3. Show that the Hurewizc homomorphism is natural.
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M8130 Algebraic topology, tutorial 11, 2020
6. 1. 2020
Exercise 4. Show that the Hurewicz homomorphism commutes with connecting homomor-phisms. It means: Let (X, A) be a pair. Show that the following diagram commutes:
nn(X,A,x0)
Hn(X,A)-
■Kn-l(A,X0) h
-tfn-l(^)
where d is the boundary homomorphism, h is the Hurewicz homomorphism and <9* is the connecting homomorphism.
S ^ dS * S I
Jb CfJ
-> a p, (s) =
M8130 Algebraic topology, tutorial 11, 2020 6. 1. 2020
Exercise 5. Show that the Hurewicz homomorphism h : nn(Sn) —> Hn(Sn) is an isomorphism.
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M8130 Algebraic topology, tutorial 11, 2020 6. 1. 2020
Exercise 6. Use Hopf fibration S1 -> S'3     S2 to compute ^(S12).
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M8130 Algebraic topology, tutorial 11, 2020 Q. 1. 2020
Exercise 7. (application) We know that deg(f) is an invariant of [Sn,Sn] = nn(Sn). Study [S2™-1, Sn] = Ti2n-i{Sn) and describe its co called Hopf invariant H(f).
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M8130 Algebraic topology, tutorial 11, 2020
6. 1. 2020
Exercise 8. What can we say in this case about Hopf inviariant for n odd and for n even?
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M8130 Algebraic topology, tutorial 11, 2020 6. 1. 2020
Exercise 9. Show that H(f) = 1 for the Hopf fibration f : S3 ->• S2.
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M8130 Algebraic topology, tutorial 11, 2020 6. 1. 2020
Exercise 10. Find a map f with Hopf invariant H(f) = 2.
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f  :   g2*-" —-y S*"     <u*cA  sO*asC =2-.
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