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