Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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V.2 Closed Surfaces 187<br />
<strong>Simplicial</strong> Approximation Theorem, there are a subdivision of I and a simplicial<br />
approximation γ ′ : I →|K| of γ that may be viewed as a path of 1-simplexes of<br />
K,say,<br />
α = {b,x0}.{x0,x1}.....{xn,a}.<br />
Let xr be the last vertex of α <strong>in</strong> L; {xr,xr+1} is then a 1-simplex that is not <strong>in</strong> L (we<br />
may assume xr �= xr+1 because a �∈ L). It follows that<br />
|L| = |L|∪|{xr,xr+1}|<br />
is a unidimensional subcomplex of |K|, strictly larger than |L| and contractible, as<br />
the union of two contractible spaces with a po<strong>in</strong>t <strong>in</strong> common. Hence, |L| is not<br />
spann<strong>in</strong>g, aga<strong>in</strong>st the hypotesis. �<br />
We now consider a spann<strong>in</strong>g tree T ⊂ G, that has the follow<strong>in</strong>g properties:<br />
(a) T conta<strong>in</strong>s all vertices of G.<br />
(b) T is a tree, <strong>in</strong> other words, |T| is contractible to a po<strong>in</strong>t.<br />
We f<strong>in</strong>ally def<strong>in</strong>e the subcomplex KT ⊂ K whose vertices are precisely those of K<br />
and whose edges are all edges of K not crossed by any side of G. S<strong>in</strong>ceT is a tree,<br />
it is easily shown that KT is connected. We see <strong>in</strong> Fig. V.16 the triangulation for the<br />
0<br />
2<br />
1<br />
3 4<br />
1<br />
5<br />
0<br />
2<br />
Fig. V.16 The tree T and the<br />
graph KT for the projective<br />
plane<br />
projective plane with the two graphs, T (<strong>in</strong> thicker l<strong>in</strong>e) and KT (<strong>in</strong> broken l<strong>in</strong>e). If<br />
we now consider the second barycentric subdivision of K, we may have two open<br />
sets U ⊃|T | and V ⊃|KT | such that<br />
(a) U ∩V = /0andU∪V = |K|<br />
•<br />
(b) U = •<br />
V<br />
as shown <strong>in</strong> Fig. V.16. 3 We note that U is homeomorphic to the disk D 2 ,s<strong>in</strong>ceT is<br />
contractible.<br />
We now consider the Euler–Po<strong>in</strong>caré characteristic χ(KT ) (<strong>in</strong> this regard, see<br />
also Exercise 6 on p. 88). We have χ(KT ) ≤ 1andχ(KT )=1 if and only if KT is<br />
3 For <strong>in</strong>stance, U may be def<strong>in</strong>ed as |T| together with the <strong>in</strong>terior of all triangles and the sides of<br />
the second barycentric subdivision of K that <strong>in</strong>tersect T.