are examples for triangulated spaces which do not admit a PL- structure. Consider an - dimensional PL- Homology-sphere . The double suspension is a topological -sphere. Choosing a triangulation obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL- manifold, because there is a vertex such that is not a sphere. A question arising with the definition is if PL-structures are always unique: Given two PL- structures for the same space , is there a there a homeomorphism which is piecewise linear with respect to both PL- structures? The assumption is similar to the Hauptvermutung and indeed there are spaces which have different PL-structures which are not equivalent. Triangulation of PL- equivalent spaces can be transformed into one another via Pachner moves: Pachner Moves Pachner moves are a way to manipulate triangulations: Let be a simplicial complex. For two simplices the Join are the points lying on straights between points in and in . Choose such that for any lying not in . A new complex , can be obtained by replacing by . This replacement is called a Pachner move. The theorem of Pachner states that whenever two triangulated manifolds are PL- equivalent, there is a series of Pachner moves transforming both into another. CW-complexes A similar but more flexible construction than simplicial complexes is the one of CW-complexes. Its construction is as follows: An - cell is the closed - dimensional unit-ball , an open -cell is its inner . Let be a topological space, let be a continuous map. The gluing is said to be obtained by gluing on an -cell. A cell complex is a union of topological spaces such that is a discrete set each is obtained from by gluing on a family of -cells. Each simplicial complex is a CW-complex, the inverse is not true. The construction of CW- complexes can be used to define cellular homology and one can show that cellular homology and simplicial homology coincide. For computational issues, it is sometimes easier to assume spaces to be CW- complexes and determine their homology via cellular decomposition, an example is the projective plane : Its construction as a CW-complex needs three cells, whereas its simplicial complex consists of 54 simplices. Other Applications Classification of manifolds By triangulating 1-dimensional manifolds, one can show that they are always homeomorphic to disjoint copies of the real line and the unit