Simplicial Structures in Topology

VI.1 Fundamental Group 207
c being the generator of π(|∂σ|,a0) (notice that
|W � σ|∩|σ| = |∂σ|
and that ∂σ is the one-dimensional simplicial complex defined by the 1-simplexes
{b0,c0}, {c0,d0},and{d0,b0}).
Since π(|σ|,a0) =0, the matter becomes much simpler: π(X,a0) derives from
π(|W � σ|,a0) together with the relation
π(iW�σ )(c)=|{b0,c0}|.|{c0,d0}|.|{d0,b0}|.
The only thing left to do now is to compute the fundamental group of |W � σ|. To
this end, we consider the barycenter b(σ) of σ and the radial projection of |D � σ|
on |∂D|; we obtain a retraction that extends to a strong deformation retraction (see
the definition given in Exercise 2 of Sect. I.2)
F : |W � σ|−→|K|
where F(|{b0,c0}|.|{c0,d0}|.|{d0,b0}|)=α. We conclude the proof by noting that
F induces an isomorphism among the homotopy groups concerned. �
Let us recalculate the fundamental group of the real projective plane without using
so many generators and relations as we did when we applied Theorem (VI.1.7).
We know that RP 2 is obtained from a disk D 2 by identifying the antipodal points of
the boundary ∂D 2 ;inotherwords,RP 2 is a pushout of the diagram
g
S 1
��
D 2
f
��
S 1
f : S 1 → S 1 , e iθ ↦→ e 2iθ .
We have only one generator (that of π(S 1 ,a0) given by the closed path α) and only
one relation α 2 ; therefore, π(RP 2 ,a0) ∼ = Z2.
VI.1.2 Polyhedra with a Given Fundamental Group
We intend to prove that, for every group G given by a finite number of generators and
relations, there exists a polyhedron whose fundamental group is G. More precisely:
(VI.1.16) Theorem. Let G = Gp(S;R) be a group with generators S = {g1,...,gm}
and relations R = {r1,...,rn}. Then there exists a two-dimensional polyhedron |K|
such that
π(|K|,a0) ∼ = G
for some vertex a0 ∈ K.