Editing Triangulation (topology) (section)

== Invariants ==
Triangulations of spaces allow assigning combinatorial invariants rising from their dedicated simplicial complexes to spaces. These are characteristics that equal for complexes that are isomorphic via a simplicial map and thus have the same combinatorial pattern.

This data might be useful to classify topological spaces up to homeomorphism but only given that the characteristics are also topological invariants, meaning, they do not depend on the chosen triangulation. For the data listed here, this is the case.<ref>{{citation|surname1=J. W. Alexander|periodical=Transactions of the American Mathematical Society|title=Combinatorial Analysis Situs|volume=28|issue=2|at=pp.&nbsp;301–329|issn=0002-9947|jstor=1989117|date=1926|doi=10.1090/S0002-9947-1926-1501346-5 |doi-access=free}}</ref> For details and the link to [[singular homology]], see topological invariance.

=== Homology ===
Via triangulation, one can assign a [[chain complex]] to topological spaces that arise from its simplicial complex and compute its ''[[simplicial homology]]''. [[Compact space|Compact]] spaces always admit finite triangulations and therefore their homology groups are [[Finitely generated abelian group|finitely generated]] and only finitely many of them do not vanish. Other data as [[Betti number|Betti-numbers]] or [[Euler characteristic]] can be derived from homology.

==== 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.

=== Topological invariance ===
To use these invariants for the classification of topological spaces up to homeomorphism one needs invariance of the characteristics regarding homeomorphism.

A famous approach to the question was at the beginning of the 20th century the attempt to show that any two triangulations of the same topological space admit a common ''subdivision''. This assumption is known as ''Hauptvermutung ('' German: Main assumption). Let <math>|\mathcal{L}|\subset \mathbb{R}^N </math>  be a simplicial complex. A complex <math> |\mathcal{L'}|\subset \mathbb{R}^N</math> is said to be a subdivision of <math>\mathcal{L}</math> iff:

* every simplex of <math>\mathcal{L'} </math> is contained in a simplex of <math>\mathcal{L} </math> and
* every simplex of <math>\mathcal{L} </math> is a finite union of simplices in <math>\mathcal{L'} </math> .<ref name=":04"/>

Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map <math>f: \mathcal{K} \rightarrow \mathcal{L}</math> between simplicial complexes is said to be piecewise linear if there is a refinement <math>\mathcal{K'}</math> of <math>\mathcal{K}</math> such that <math>f</math> is piecewise linear on each simplex of <math>\mathcal{K}</math>. Two complexes that correspond to another via piecewise linear bijection are said to be combinatorial isomorphic. In particular, two complexes that have a common refinement are combinatorially equivalent. Homology groups are invariant to combinatorial equivalence and therefore the Hauptvermutung would give the topological invariance of simplicial homology groups. In 1918, Alexander introduced the concept of singular homology. Henceforth, most of the invariants arising from triangulation were replaced by invariants arising from singular homology. For those new invariants, it can be shown that they were invariant regarding homeomorphism and even regarding [[homotopy equivalence]].<ref name=":32">{{citation|surname1=Allen Hatcher|title=Algebraic Topologie|publisher=Cambridge University Press|publication-place=Cambridge/New York/Melbourne|at=p.&nbsp;110|isbn=0-521-79160--X|date=2006
}}</ref> Furthermore it was shown that singular and simplicial homology groups coincide.<ref name=":32"/> This workaround has shown the invariance of the data to homeomorphism. Hauptvermutung lost in importance but it was initial for a new branch in topology: The ''piecewise linear topology'' (short PL-topology).<ref>{{cite web|title=One the Hauptvermutung|periodical=The Hauptvermutung Book|publisher=|url=https://www.maths.ed.ac.uk/~v1ranick/books/haupt.pdf|url-status=|format=|access-date=|archive-url=|archive-date=|last=A.A.Ranicki|date=|year=|pages=|quote=}}</ref>