Simplicial Structures in Topology

218 VI Homotopy Groups
4. R ′ α : I 2 −→ I 2
(∀(s,t) ∈ I 2 ) R ′ � � �
1 1
α(s,t)= , + �ρα s −
2 2
1
�
1
,t −
2 2
The function �ρα may be geometrically described as follows: given a vector �v =
(y1,y2) ∈ R 2 ,letr = |(y1,y2)|; the point (y1,y2) belongs to the boundary of the
square Qr with vertices (r,r), (−r,r), (−r,−r), and(r,−r); we rotate the half-line
(0,0) , (y1,y2) through an angle α π 2 , ending up with a half-line that crosses Qr at
the point �ρα(y1,y2). With this description in mind, we shall say that �ρα is a square
rotation of R 2 with center at (0,0) and through the angle α π 2 . It follows that R′ α is a
square rotation of I 2 about ( 1 2 , 1 2 ) and through the angle α π 2 ; therefore, R′ α induces
amap
Rα : ΣΣS n−1 −→ ΣΣS n−1
The homotopy
(∀s ∧t ∧ x ∈ ΣΣS n−1 ) Rα(s ∧t ∧ x)=R ′ α (s ∧t) ∧ x.
F : Σ 2 S n−1 × I −→ Σ 2 S n−1 ∨ Σ 2 S n−1
that we seek is the following composition:
(∀u ∈ I) F(−,u)=(R−u ∨ R−u){h(Σνn)}Ru. �
(VI.3.6) Remark. Notice that by iteration
νn+1 ∼ (hΣ) n ν1.
We recall that the symbol ΩY indicates the space of paths of a based space (Y,y0)
(see p. 32).
(VI.3.7) Theorem. For every n ≥ 2 and every (Y,y0) ∈ Top ∗, the groups πn(Y,y0)
and πn−1(ΩY,cy0 ) are isomorphic.
Proof. We know that every based map
has an adjoint map
f : S n ∼ = ΣS n−1 → Y
f : S n−1 → ΩY
such that, for every x ∈ S n−1 and every t ∈ I,
{ f (x)}(t)= f (t ∧ x).
In Sect. I.2, we have seen that the function
Φ : M∗(ΣX,Y ) → M∗(X,ΩY) , f ↦→ f