it admits a triangulation, namely the homeomorphism . Restricting to yields a homeomorphism . The torus admits a triangulation. To see this, consider the torus as a square where the parallel faces are glued together. This square can be triangulated as shown below: The projective plane admits a triangulation (see CW-complexes) One can show that differentiable manifolds admit triangulations. Invariants Triangulations of spaces allow assigning combinatorial invariants rising from their dedicated simplicial complexes to spaces. These are characteristics that equal for complexes that are isomorphic via a simplicial map and thus have the same combinatorial pattern. This data might be useful to classify topological spaces up to homeomorphism but only given that the characteristics are also topological invariants, meaning, they do not depend on the chosen triangulation. For the data listed here, this is the case. For details and the link to singular homology, see topological invariance Homology Via triangulation, one can assign a chain complex to topological spaces that arise from its simplicial complex and compute its simplicial homology. Compact spaces always admit finite triangulations and therefore their homology groups are finitely generated and only finitely many of them do not vanish. Other data as Betti- Numbers or Euler characteristic can be derived from homology. Betti- numbers and Euler-characteristics Let be a finite simplicial complex. The - th Betti- number is defined to be the rank of the - th simplicial homology- group of the spaces. These numbers encode geometric properties of the spaces: The Betti- Number for instance represents the number of connected components. For a triangulated, closed orientable surfaces , holds where denotes the genus of the surface: Therefore its first Betti- number represents the doubled number of handles of the surface. With the comments above, for compact spaces all Betti- numbers are finite and almost all are zero. Therefore, one can form their alternating sum which is called the Euler Charakteristik of the complex, a catchy topological invariant. Topological invariance To use these invariants for the classification of topological spaces up to homeomorphism one needs invariance of the characteristics regarding homeomorphism. A famous approach to the question was at the beginning of the 20th century the attempt to show that any two triangulations of the same topological space admit a common subdivision. This assumption is known as Hauptvermutung ( German: Main assumption). Let be a simplicial complex. A complex is said to be a subdivision of