Simplicial Structures in Topology

VI.1 Fundamental Group 203
are reduced to the identity of the fundamental group; besides,
g02 = g04 = g05 = g15 = g23 = g25
and so we have only one generator, let us say, g15, but with the property (g15) 2 = 1
in π(RP 2 ,0).
Theorem (VI.1.7) enables us to compute the fundamental group of a pathconnected
polyhedron, but at a price: indeed, the number of generators and relations
may be excessively large. So, we try to obtain results in a more economical way;
with this in mind, let us consider a few things.
In Sect. I.2, we have proved that the category of groups is closed by pushouts.
We now review this assertion. Let G1 and G2 be two groups given by generators
and relations
G1 = Gp(S1,R1) and G2 = Gp(S2,R2);
We now consider the homomorphisms f : G → G1 and g: G → G2, and form
the pushout of the pair ( f ,g) to obtain the group G1 ∗ f ,g G2. This group is actually
isomorphic to the group presented as Gp(S1 ∪ S2;R1 ∪ R2 ∪ R f ,g), where
R f ,g = { f (x)g(x) −1 |x ∈ G}. By definition of pushout, the group G1 ∗ f ,g G2 has
the following properties:
1. There exist two homomorphisms
g: G1 −→ G1 ∗ f ,g G2
f : G2 −→ G1 ∗ f ,g G2
such that fg= gf
2. For every group H and homomorphisms
h: G1 → H and k : G2 → H
such that hf = kg, there exists a unique homomorphism
such that
ℓ: G1 ∗ f ,g G2 −→ H
ℓ f = k and ℓg = h
Specifically, if G = 0then f = g = 0, R f ,g = 0, and the group
G1 ∗0,0 G2 := Gp(S1 ∪ S2;R1 ∪ R2)
(also denoted by the symbol G1 ∗ G2) is called free product of G1 and G2.