Simplicial Structures in Topology

196 VI Homotopy Groups
and define the map (called comultiplication in S1 )
ν : S 1 → S 1 ∨ S 1 , e 2πti �
(e0,e
↦→
2π2ti ) 0 ≤ t ≤ 1 2
(e2π(2t−1)i ,e0) 1
2 ≤ t ≤ 1;
besides, let us define the folding map σ : S 1 ∨ S 1 → S 1 :
(∀0 ≤ t ≤ 1)(e0,e 2tiπ ) ↦→ e 2tiπ , (e 2tiπ ,e0) ↦→ e 2tiπ .
By these definitions, one can easily see that for every point e 2πti ∈ S 1
( f ∗ g)(e 2πti )=σ( f ∨ g)ν(e 2πti );
moreover, we note that if f ′ ∼ f and g ′ ∼ g, thenσ( f ′ ∨ g ′ )ν is homotopic to
σ( f ∨ g)ν. Finally, for any based space (Y,y0), we define the function (called multiplication)
[S 1 ,Y ]∗ × [S 1 ,Y ]∗ −→ [S 1 ,Y ]∗
(∀[ f ],[g] ∈ [S 1 ,Y ]∗)[f ] ν
× [g] :=[σ( f ∨ g)ν].
(VI.1.1) Theorem. The set [S 1 ,Y ]∗ with the multiplication ν
× is a group whose unit
element is the homotopy class of the constant function c: S 1 → Y (it takes each
point of S 1 to y0); besides, the inverse of [ f ] is the homotopy class of the based map
h: S 1 → Y defined, for each e 2πti ∈ S 1 , by the formula h(e 2πti )= f (e 2π(1−t)i ). 1
Proof. We first prove that the function ν : S 1 → S 1 ∨ S 1 is associative (up to homotopy),
in other words, that the diagram
ν
S 1 ν ��
��
S 1 × S 1
S 1 ∨ S 1
1 ∨ ν
��
ν ∨ 1 ��
S 1 ∨ S 1 ∨ S 1
commutes up to homotopy (that is to say (1 ∨ ν)ν ∼ (ν ∨ 1)ν). In fact, for every
e2πti ∈ S1 ,
(1 ∨ ν)ν(e 2πti ⎧
⎨ (e0,e0,e
)=
⎩
2π2ti ) 0 ≤ t ≤ 1 2
(e0,e2π2(2t−1)i ,e0) 1 2 ≤ t ≤ 3 4
(e0,e0,e 2π(4t−3)ti ) 3
4 ≤ t ≤ 1
(ν ∨ 1)ν(e 2πti ⎧
⎨ (e0,e0,e
)=
⎩
2π4ti ) 0 ≤ t ≤ 1 4
(e0,e2π(4t−1)i ,e0) 1 4 ≤ t ≤ 1 2
(e2π(2t−1)i ,e0,e0) 1 2 ≤ t ≤ 1
1 Intuitively, the loop h is the loop f , but traveled in the opposite direction.