Simplicial Structures in Topology

208 VI Homotopy Groups
Proof. For each generator gi, we take a circle S1 i with a base point ei ∈ S1 i , i =
1,...,m. We consider the set
Y = S 1 1 ∨ S 1 2 ∨ ...∨ S 1 m
of all m-tuples (x1,...,xm) ∈ S 1 1 × ...S1 m with no more than one coordinate xi different
from its base point ei; wegiveY the base point a0 =(e1,...,em) and the
topology induced by the topology of the product space S 1 1 × ...× S1 m.Wepausehere
to interpret Y in another way. We consider the pushout
{x}
��
S 1 2
i2
i1 ��
i1
(with i j(x) =e j, j = 1,2); note that Z ∼ = S1 1 ∨ S1 2 and, by induction, also S1 1 ∨ S1 2 ∨
...∨ S1 m may be viewed as the pushout space of an appropriate diagram. Besides,
we observe that, due to Corollary (VI.1.14), π(S1 1 ∨ S1 2 ,a0) is a free group with
two generators and, in general, π(S1 1 ∨ S1 2 ∨ ...∨ S1 m,a0) is a free group with m
generators.
We now return to our theorem. A relation, let us say, r j is a word
with εi = ±1; we define a function
i2
S 1 1
��
��
Z
r j = b ε1
i1 ...bεp
ip
f j : S 1 −→ Y,
corresponding to the word r j, as follows: since the word r j has p letters, we divide
S 1 into p equal parts; the qth arc is completely wrapped around the component S 1 iq
of Y , clockwise if εq =+1 and counter-clockwise if εq = −1. Analytically, such a
function is described as follows:
f j(e iθ )=
�
e i(pθ−2(q−1)π) if εq = 1
e i(2qπ−pθ) if εq = −1,
where 2(q−1)π/p ≤ θ ≤ 2qπ/p, 1 ≤ q ≤ p (look up the function f : S 1 → S 1 used
for computing the fundamental group of the real projective plane, just before this
subsection).