Simplicial Structures in Topology

VI.1 Fundamental Group 199
Proof. Let us suppose that φ λ = φμ. Then, for every [ f ] ∈ π(Y,y0),
λ −1 ∗ f ∗ λ ∼ μ −1 ∗ f ∗ μ
λ −1 ∗ f ∗ λ ∗ μ −1 ∼ μ −1 ∗ f
(μ ∗ λ −1 ) ∗ f ∗ (λ ∗ μ −1 ) ∼ f ;
since (μ ∗ λ −1 ) −1 = λ ∗ μ −1 ,wehavethat
and therefore
f ∗ (λ ∗ μ −1 ) ∼ (λ ∗ μ −1 ) ∗ f
[λ ∗ μ −1 ] ∈ Zπ(Y,y0).
Proving the converse is equally easy. �
(VI.1.6) Corollary. The isomorphism φ λ : π(Y,y0) → π(Y,y1) associated with a
path λ : I → Yfromy0 to y1 does not depend on λ when π(Y,y0) is Abelian.
VI.1.1 Fundamental Groups of Polyhedra
There are no fixed rules for computing the fundamental group of an arbitrary-based
space; there is, however, a practical and efficient method for computing the fundamental
group of a polyhedron based at a vertex. The reader is advised to review
the material on simplicial complexes and their geometric realizations, also because
of the notation. We recall that, by Lemma (V.2.6) on p. 186, there exists a onedimensional
subcomplex of each connected finite simplicial complex K that contains
all vertices and whose geometric realization is contractible; such a subcomplex
is a spanning tree and is such that:
1. dim(L)=1.
2. L contains all vertices of K (that is to say, X = Y).
3. The polyhedron |L| is contractible to a point.
(VI.1.7) Theorem. Let K =(X,Φ) be a connected simplicial complex with a fixed
vertex a0 and let L =(Y,Θ) be a spanning tree in K. Let a symbol gij be associated
with every 1-simplex {ai,a j}∈Φ and let S be the set of all symbols gij obtained in
this manner; let
R = {gij|{ai,a j}∈Θ}∪{gijg jkgki |{ai,aj,ak}∈Φ}.
Then, π(|K|,a0) is isomorphic to the group Gp(S;R) generated by the set S with the
relations R.