Editing Triangulation (topology) (section)

=== 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>