Editing Triangulation (topology) (section)

=== 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.