Simplicial Structures in Topology

VI.3 Homotopy Groups 217
such that the following diagram is commutative:
| •
1 •
| σn+1|
σn+1|×{0}
× ι0
��
| •
σn+1|×I
ι × 1 {0}
ι × 1 {0}
��
1 •
| σn+1| ��
G
|σn+1|×{0}
��
× ι0
��
|σn+1|×I
���
���
��
H ���
c
��
��
��
Y
The restriction of H to |σn+1|×1 = |σn+1| is the desired extension of f . �
Our next lemma is important to the proof of the two theorems that follow it.
(VI.3.5) Lemma. For every n ≥ 1, the function
such that, for every t ∈ I and x ∈ S n ,
h: Σ(S n ∨ S n ) −→ ΣS n ∨ ΣS n
h(t ∧ (x,e0)) = (t ∧ x,e0), h(t ∧ (e0,x)) = (e0,t ∧ x),
is a homeomorphism; moreover, the maps νn+1 and (hΣ)νn are homotopic.
Proof. The definitions are such that, for every t ∧ s ∧ x ∈ Σ 2Sn−1 ∼ = ΣSn ,
�
(t ∧ 2s ∧ x,e0) 0 ≤ s ≤
h(Σνn)(t ∧ s ∧ x)=
1 2 ,
(e0,t ∧ (2s − 1) ∧ x) 1
2 ≤ s ≤ 1,
�
(2t ∧ s ∧ x,e0) 0 ≤ s ≤
νn+1(t ∧ s ∧ x)=
1
2 ,
(e0,(2t − 1) ∧ s ∧ x) 1
2 ≤ s ≤ 1.
We now define the following functions:
1. ||: R2 −→ R , |(y1,y2)| = max(|y1|,|y2|), forevery(y1,y2) ∈ R2 2. For every α ∈ [−1,1],
ρα : R 2 −→ R 2
(∀(y1,y2) ∈ R 2 �
π cosα
) ρα(y1,y2)= 2 −sinα π 2
sinα π 2 cosα π ��
y1
2
y2
3. �ρα : R 2 −→ R 2
(∀(y1,y2) ∈ R 2 ) �ρα(y1,y2)= |(y1,y2)|
|ρα(y1,y2)| ρα(y1,y2)
�