Simplicial Structures in Topology

202 VI Homotopy Groups
(VI.1.10) Example (π(S 1 ∨ S 1 ) is the free group with two generators). This group
is not Abelian. It is an immediate consequence from Theorem (VI.1.7), whenwe
triangulate each circle as the boundary of a triangle.
(VI.1.11) Example (π(T 2 ) ∼ = Z×Z). We consider the triangulation of a torus with
9 vertices, 27 edges, and 18 faces (see figure in Sect. III.5.1). The spanning tree |L|
is given by the geometric realizations of the following 1-simplexes (to simplify the
notation, we omit curly brackets and commas between two vertices):
01,12,23,34,46,65,57,78
Sixteen of the twenty seven generators gij become the identity in π(T 2 ,0):
g01,g05,g12,g15,g16,g23,g26,g34,g36,g46,g48,g56,g57,g58,g68,g78;
but there are also the relations
g02g28g80 and g17g74g41.
All this leads us to the conclusion that we only have two generators in the fundamental
group that are not reduced to the identity: g02 and g28; besides,
g02g28(g02) (−1) (g28) (−1) = 1
in π(T 2 ,0) which is, therefore, isomorphic to Z ⊕ Z. 2
0
2
1
3 4
5
1
2
0
Fig. VI.1 A triangulation of
the real projective plane
(VI.1.12) Example (π(RP 2 ) ∼ = Z2). Let us consider the triangulation of the real
projective plane with 6 vertices, 15 edges, and 10 faces, as in Sect. III.5. Thespanning
tree |L| is given by the geometric realizations of the following 1-simplexes:
01,12,24,43,35, as shown in Fig. VI.1. In this example, the generators
g01,g03,g12,g13,g14,g24,g34,g35,g45
2 Another way to prove this result is as follows: if X and Y are topological spaces, then the
fundamental group of the product X × Y coincides with the direct product of the fundamental
groups π(X) and π(Y) (see Exercise 3 on p. 210).