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