Simplicial Structures in Topology

230 VI Homotopy Groups
f : I n
g: I n
q
q
��
n
S
��
n
S
f
g
��
Y
��
Y.
On the other hand, we define the map θ : I n → I n ∨ I n such that, for every
we have
(x1,...,xn) ∈ I n ,
�
(∗,(2x1,...,xn)) 0 ≤ x1 ≤
θ(x1,...,xn)=
1 2
((2x1 − 1,...,xn),∗) 1 2 ≤ x1 ≤ 1;
we then notice that the following diagram is commutative:
θ
I n
��
I n ∨ I n
q
q ∨ q
By directly applying the definitions, we have
Exercises
��
S n ≡ ΣS n−1
νn
��
��
n n
S ∨ S .
(σ( f ∨ g)νn)q = σ( f ∨ g)(q ∨ q)θ =
= σ( f ∨ g)θ = f ∗ g .�
1. Let X be any based space. Prove that the suspension ΣX of X is a space with an
associative comultiplication.
2. Prove that for every (X,x0),(Y,y0) ∈ Top ∗, the set of based homotopy classes
[ΣX,ΩY ]∗ is an Abelian group.
3. Let f : A → B and g: Y → B be two given maps; take the space
X = {(a,y) ∈ A ×Y| f (a)=g(y)}
with the projections pr 1 : X → A and pr 2 : X → B. Prove that f pr 1 = gpr 2 . Furthermore,
prove that, for every topological space Z and any maps h: Z → Y and
k : Z → A such that fk= gh, there exists a unique map ℓ: Z → X such that pr 1 ℓ =
pr 2 ℓ. This is an example of pullback, the pushout dual in Top, which was defined in
Sect. I.2. This situation is depicted by the following commutative diagram: