Editing Triangulation (topology) (section)

== Hauptvermutung ==
The Hauptvermutung (''German for main conjecture'') states that two triangulations always admit a common subdivision. Originally, its purpose was to prove invariance of combinatorial invariants regarding homeomorphisms. The assumption that such subdivisions exist in general is intuitive, as subdivision are easy to construct for simple spaces, for instance for low dimensional manifolds. Indeed the assumption was proven for manifolds of dimension <math>\leq 3</math> and for differentiable manifolds but it was disproved in general:<ref name=":42">{{citation|surname1=John Milnor|periodical=The Annals of Mathematics|title=Two Complexes Which are Homeomorphic But Combinatorially Distinct|year=1961 |volume=74|issue=3|at=p.&nbsp;575|issn=0003-486X|doi=10.2307/1970299
|jstor=1970299 }}</ref>  An important tool to show that triangulations do not admit a common subdivision, that is, their underlying complexes are not combinatorially isomorphic is the combinatorial invariant of Reidemeister torsion.

=== Reidemeister torsion ===
To disprove the Hauptvermutung it is helpful to use combinatorial invariants which are not topological invariants. A famous example is Reidemeister torsion. It can be assigned to a tuple <math>(K,L)</math> of CW-complexes: If <math>L = \emptyset</math> this characteristic will be a topological invariant but if <math>L \neq \emptyset</math> in general not. An approach to Hauptvermutung was to find homeomorphic spaces with different values of Reidemeister torsion. This invariant was used initially to classify lens-spaces and first counterexamples to the Hauptvermutung were built based on lens-spaces:<ref name=":42"/>

=== Classification of lens spaces ===
In its original formulation, [[lens spaces]] are 3-manifolds, constructed as quotient spaces of the 3-sphere: Let <math>p, q</math> be natural numbers, such that  <math>p, q</math> are coprime. The lens space <math>L(p,q)</math> is defined to be the orbit space of the [[free group action]]

:<math>\Z/p\Z\times S^{3}\to S^{3}</math>
:<math>(k,(z_1,z_2)) \mapsto (z_1 \cdot e^{2\pi i k/p}, z_2 \cdot e^{2\pi i kq/p} )</math>.

For different tuples <math>(p, q)</math>, lens spaces will be homotopy equivalent but not homeomorphic.  Therefore they can't be distinguished with the help of classical invariants as the fundamental group but by the use of Reidemeister torsion.

Two lens spaces <math>L(p,q_1), L(p,q_2)</math> are homeomorphic, if and only if <math>q_1 \equiv \pm q_2^{\pm 1} \pmod{p}  </math>.<ref>{{citation|surname1=Marshall M. Cohen|periodical=Graduate Texts in Mathematics|title=A Course in Simple-Homotopy Theory|series=Graduate Texts in Mathematics |issn=0072-5285|date=1973|volume=10 |doi=10.1007/978-1-4684-9372-6
|isbn=978-0-387-90055-1 }}</ref> This is the case if and only if two lens spaces are ''simple homotopy equivalent''. The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces <math>L'_1, L'_2</math> derived from non-homeomorphic lens spaces <math>L(p,q_1), L(p,q_2)</math> having different Reidemeister torsion.  Suppose further that the modification into <math>L'_1, L'_2</math> does not affect Reidemeister torsion but such that after modification <math>L'_1</math> and <math>L'_2</math> are homeomorphic. The resulting spaces will disprove the Hauptvermutung.