Simplicial Structures in Topology

VI.1 Fundamental Group 197
The function
H : S 1 × I −→ S 1 ∨ S 1 ∨ S 1 ,
defined for every (e 2πti ,s) ∈ S 1 × I by the formula
H(e 2πti ⎧
⎨ (e0,e0,e
,s)=
⎩
2π(2s+2)ti ) 0 ≤ t ≤ 2−s
4
(e0,e2π[4t−2(1− s 2 )]i 2−s 3−s
,e0) 4 ≤ t ≤ 4
≤ t ≤ 1,
(e 2π[4(1− s 2 )(t−1)+1]i ,e0,e0) 3−s
4
is the desired homotopy.
We now prove that the multiplication ν
× is associative. We have to prove that
(∀ f ,g,h ∈ Top ∗(S 1 ,Y)) σ(σ( f ∨ g)ν ∨ h)ν ∼ σ( f ∨ σ(g ∨ h)ν)ν.
In fact, the associativity of ν implies
on the other hand,
σ(1Y ∨ σ)( f ∨ g ∨ h)(1 ∨ ν)ν ∼ σ(σ ∨ 1Y )( f ∨ g ∨ h)(ν ∨ 1)ν;
σ(1Y ∨ σ)( f ∨ g ∨ h)(1 ∨ ν)ν = σ( f ∨ σ(g ∨ h)ν)ν
σ(σ ∨ 1Y )( f ∨ g ∨ h)(ν ∨ 1)ν = σ(σ( f ∨ g)ν ∨ h)ν.
The homotopy class [c] of the constant function at the base point y0 ∈ Y is the
identity element of [S1 ,Y ]∗. We first prove that, if i : S1 ∨ S1 → S1 × S1 is the inclusion
map and Δ : S1 → S1 × S1 is the diagonal function e2πti ↦→ (e2πti ,e2πti ),then
iν ∼ Δ; in fact, this assertion is ensured by the homotopy H : S1 × I → S1 × S1 H(e 2πti �
(e2πtsi ,e2π(t(2−s)i 1
) 0 ≤ t ≤
,s)=
2
(e2π[2t−1+s(1−t)]i ,e2π(st+1−s)i ) 1 2 ≤ t ≤ 1.
We then conclude that, for every based map f : S 1 → Y ,
σ( f ∨ c)ν = σ( f × c)iν ∼ σ( f × c)Δ = f
σ(c ∨ f )ν = σ(c × f )iν ∼ σ(c × f )Δ = f .
Finally, we prove that every [ f ] ∈ [S1 ,Y ]∗ has an inverse. Indeed, for each e2πti ,we
define h: S1 → Y by h(e2πti )= f (e2π(1−t)i ) and note that
σ( f ∨ h)ν(e 2πti �
f (e2π2ti 1 ) 0 ≤ t ≤
)=
2
h(e2π(2t−1)i ) 1
2 ≤ t ≤ 1;
the function
H(e 2πti ⎧
y0 ⎪⎨
0 ≤ t ≤
,s)=
⎪⎩
s 2
f (e2π(2t−s)i s ) 2 ≤ t ≤ 1 2
f (e2π(2−2t−s)i ) 1 2−s
2 ≤ t ≤ 2
≤ t ≤ 1
y0
2−s
2