Problem Set 1
Due Thursday, March 3
Problem 1. S ∞ ' ∗ (Hint: consider f : S ∞ → S ∞ , (x1 , x2 , . . . ) 7→ (0, x1 , x2 , . . . )).
Problem 2. Show that S n−1 ,→ Dn is a cofibration. Deduce that any relative CW complex
A ,→ X is a cofibration.
Problem 3. Let i : A ,→ X and j : B ,→ Y be cofibrations. Show that i×j : A×B ,→ X ×Y
is a cofibration (Hint: assume first that j is the identity).
Problem 4. Prove the following statements.
(a) If A is contractible and A ,→ X is a cofibration, then X → X/A is a homotopy
equivalence.
(b) If i : A ,→ X is a cofibration, then Cone(i) → X/A is a homotopy equivalence.
Problem 5. (Optional) Recall that a pointed space (X, x0 ) is well pointed if the inclusion
{x0 } ,→ X is a cofibration. Let f : X → Y be a pointed map with X and Y well pointed.
Show that the canonical map Cone(f ) → Cone∗ (f ) is a homotopy equivalence.
Hint: One approach is to first show that Cyl(f ) → Cyl∗ (f ) is a homotopy equivalence and
then apply Problem 4 (b). The assumption that X and Y are well pointed is used to show
that X ,→ Cyl∗ (f ) is a cofibration.
Problem 6. Let A be a subspace of X and let x0 ∈ A. For n ≥ 1, define the relative
homotopy group πn (X, A, x0 ) as follows:
πn (X, A, x0 ) = [(Dn , S n−1 , ∗), (X, A, x0 )].
It is only a pointed set for n = 1. For n ≥ 2, the pinch map defines a group structure.
(a) Let F be the homotopy fiber of the inclusion i : A ,→ X at x0 . Show that there is a
bijection πn (X, A, x0 ) ∼
= πn−1 (F, x0 ) and convince yourself that it is an isomorphism of
groups when n ≥ 2.
The long exact sequence for the fiber sequence F → A → X can therefore be written as
∂
i
φ
∗
· · · → πn+1 (X, A, x0 ) → πn (A, x0 ) −
→
πn (X, x0 ) → πn (X, A, x0 ) → · · · .
(b) Identify the maps ∂ and φ in terms of the definition of πn (X, A, x0 ).
Problem 7. The Hopf fibration η : S 3 → S 2 is the restriction to S 3 of the canonical
projection C2 \ 0 → CP 1 = S 2 .
(a) Show that η is a fiber bundle with fiber S 1 .
(b) Deduce that π2 (S 2 ) ∼
= Z.
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Later, we will be able to deduce from this that πn (S n ) ∼
= Z for all higher n. Moreover, one
n
n
can show that πn (S ) is generated by the identity of S .
(c) Assuming this, show that π3 (S 2 ) = Zη.
(d) Devise quaternionic and octonionic versions of the Hopf map, called ν and σ. Argue
that ν and σ have infinite order in the homotopy groups π7 (S 4 ) and π15 (S 8 ), respectively.
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