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