are 3-manifolds, constructed as quotient spaces of the 3-sphere: Let be natural numbers, such that are coprime . The lens space is defined to be the orbit space of the free group action . For different tuples , 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 are homeomorphic, if and only if . This is the case iff two Lens spaces are simple-homotopy-equivalent. The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces derived from non-homeomorphic Lens spaces having different Reidemeister torsion. Suppose further that the modification into does not affect Reidemeister torsion but such that after modification and are homeomorphic. The resulting spaces will disprove the Hauptvermutung. Existence of triangulation Besides the question of concrete triangulations for computational issues, there are statements about spaces that are easier to prove given that they are simplicial complexes. Especially manifolds are of interest. Topological manifolds of dimension are always triangulable but there are non-triangulable manifolds for dimension , for arbitrary but greater than three. Further, differentiable manifolds always admit triangulations. PL- Structures Manifolds are an important class of spaces. It is natural to require them not only to be triangulable but moreover to admit a piecewise linear atlas, a PL- structure: Let be a simplicial complex such that every point admits an open neighborhood such that there is a triangulation of and a piecewise linear homeomorphism . Then is said to be a piecewise linear (PL) manifold of dimension and the triangulation together with the PL- atlas is said to be a PL- structure on . An important lemma is the following: Let be a topological space. It is equivalent is an -dimensional manifold and admits a PL- structure. There is a triangulation of such that the link of each vertex is an sphere. For each triangulation of the link of each vertex is an sphere. The equivalence of the second and the third statement is because that the link of a vertex is independent of the chosen triangulation up to combinatorial isomorphism. One can show that differentiable manifolds admit a PL- structure as well as manifolds of dimension . Counterexamples for the triangulation conjecture are counterexamples for the conjecture of the existence of PL- structure of course. Moreover, there