iff: every simplex of is contained in a simplex of and every simplex of is a finite union of simplices in . Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map between simplicial complexes is said to be piecewise linear if there is a refinement of such that is piecewise linear on each simplex of . 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. Furthermore it was shown that singular and simplicial homology groups coincide. 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 (short PL- topology) topology examines topological properties of topological spaces. 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 and for differentiable manifolds but it was disproved in general: An important tool to show that triangulations do not admit a common subdivision. i. e 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 of CW- complexes: If this characteristic will be a topological invariant but if 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: Classification of lens- spaces In its original formulation, Lens spaces