Editing Triangulation (topology) (section)

==== Betti-numbers and Euler-characteristics ====
Let <math>|\mathcal{S}|</math> be a finite simplicial complex. The <math>n</math>-th Betti-number <math>b_n(\mathcal{S})</math> is defined to be the [[Rank of an abelian group|rank]] of the <math>n</math>-th simplicial homology group of the spaces. These numbers encode geometric properties of the spaces: The Betti-number <math>b_0(\mathcal{S})</math> for instance represents the number of [[Connected space|connected]] components. For a triangulated, closed [[Orientability|orientable]] [[Surface (mathematics)|surfaces]] <math>F</math>, <math>b_1(F)= 2g</math> holds where <math>g</math> denotes the [[Genus (mathematics)|genus]] of the surface: Therefore its first Betti-number represents the doubled number of [[Handle decomposition|handles]] of the surface.<ref>{{citation|surname1=R. Stöcker, H. Zieschang|title=Algebraische Topologie|edition=2. überarbeitete|publisher=B.G.Teubner|publication-place=Stuttgart|at=p.&nbsp;270|isbn=3-519-12226-X|date=1994|language=German
}}</ref>

With the comments above, for compact spaces all Betti-numbers are finite and almost all are zero. Therefore, one can form their alternating sum

: <math>\sum_{k=0}^{\infty} (-1)^{k}b_k(\mathcal{S})</math>

which is called the ''Euler characteristic'' of the complex, a catchy topological invariant.