Simplicial Structures in Topology

V.2 Closed Surfaces 187
Simplicial Approximation Theorem, there are a subdivision of I and a simplicial
approximation γ ′ : I →|K| of γ that may be viewed as a path of 1-simplexes of
K,say,
α = {b,x0}.{x0,x1}.....{xn,a}.
Let xr be the last vertex of α in L; {xr,xr+1} is then a 1-simplex that is not in L (we
may assume xr �= xr+1 because a �∈ L). It follows that
|L| = |L|∪|{xr,xr+1}|
is a unidimensional subcomplex of |K|, strictly larger than |L| and contractible, as
the union of two contractible spaces with a point in common. Hence, |L| is not
spanning, against the hypotesis. �
We now consider a spanning tree T ⊂ G, that has the following properties:
(a) T contains all vertices of G.
(b) T is a tree, in other words, |T| is contractible to a point.
We finally define the subcomplex KT ⊂ K whose vertices are precisely those of K
and whose edges are all edges of K not crossed by any side of G. SinceT is a tree,
it is easily shown that KT is connected. We see in Fig. V.16 the triangulation for the
0
2
1
3 4
1
5
0
2
Fig. V.16 The tree T and the
graph KT for the projective
plane
projective plane with the two graphs, T (in thicker line) and KT (in broken line). If
we now consider the second barycentric subdivision of K, we may have two open
sets U ⊃|T | and V ⊃|KT | such that
(a) U ∩V = /0andU∪V = |K|
•
(b) U = •
V
as shown in Fig. V.16. 3 We note that U is homeomorphic to the disk D 2 ,sinceT is
contractible.
We now consider the Euler–Poincaré characteristic χ(KT ) (in this regard, see
also Exercise 6 on p. 88). We have χ(KT ) ≤ 1andχ(KT )=1 if and only if KT is
3 For instance, U may be defined as |T| together with the interior of all triangles and the sides of
the second barycentric subdivision of K that intersect T.