Simplicial Structures in Topology

VI.3 Homotopy Groups 215
for every n ≥ 2 and for every (Y,y0) ∈ Top ∗; also in this case, the definition does
not depend on the representatives of the homotopy classes [ f ] and [g].
(VI.3.3) Theorem. The set [S n ,Y]∗ with the multiplication νn
× is a group.
Proof. The associativity of νn
× stems from νn being homotopy associative (see
Lemma (VI.3.1)): indeed, for every f ,g,h ∈ Top ∗((S n ,e0),(Y,y0)),
σ(1Y ∨ σ)( f ∨ g ∨ h)(1S n ∨ νn)νn ∼ σ(σ ∨ 1Y )( f ∨ g ∨ h)(νn ∨ 1S n)νn
and the equalities
σ(1Y ∨ σ)( f ∨ g ∨ h)(1S n ∨ νn)νn = σ( f ∨ σ(g ∨ h)νn)νn,
σ(σ ∨ 1Y )( f ∨ g ∨ h)(νn ∨ 1S n)νn = σ(σ( f ∨ g)νn ∨ h)νn
are also valid.
The homotopy class [c] of the constant map
c: S n → Y , t ∧ x ↦→ y0
is the identity element for the multiplication defined in πn(Y,y0): in fact, for every
f ∈ Top ∗((S n ,e0),(Y,y0)) andbyLemma(VI.3.2), the following homotopies:
σ( f ∨ c)νn = σ( f × c)ινn ∼ σ( f × c)Δ = f ,
σ(c ∨ f )νn = σ(c × f )ινn ∼ σ(c × f )Δ = f
hold true.
As for the inverses, we proceed in the following manner. Let [ f ] ∈ πn(Y,y0) be
an arbitrarily given element. We define
note that, for every t ∧ x ∈ S n ,
h: S n → Y , t ∧ x ↦→ f ((1 − t) ∧ x);
� 1
f (2t ∧ x) 0 ≤ t ≤
σ( f ∨ h)νn(t ∧ x)=
2
h((2t − 1) ∧ x) 1 2 ≤ t ≤ 1.
The homotopy
⎧
y0 ⎪⎨
0 ≤ t ≤
H(t ∧ x,s)=
⎪⎩
s
2
s
f ((2t − s) ∧ x) 2 ≤ t ≤ 1 2
f ((2 − 2t − s) ∧ x) 1 2−s
2 ≤ t ≤ 2
≤ t ≤ 1
y0
shows that [h] is the right inverse of [ f ]; similarly, one proves that [h] is also the left
inverse of [ f ]. �
2−s
2