sphere . Moreover, surfaces, i.e. 2-manifolds, can be classified completely: Let be a compact surface. If is orientable, it is homeomorphic to a 2-sphere with tori of dimension attached, for some . If is not orientable, it is homeomorphic to a Klein Bottle with tori of dimension attached, for some . To prove this theorem one constructs a fundamental polygon of the surface: This can be done by using the simplicial structure obtained by the triangulation. Maps on simplicial complexes Giving spaces the structure of a simplicial structure might help to understand maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem: Simplicial approximation Let , be abstract simplicial complexes above sets , . A simplicial map is a function which maps each simplex in onto a simplex in . By affin-linear extension on the simplices, induces a map between the geometric realizations of the complexes. Each point in a geometric complex lies in the inner of exactly one simplex, its support. Consider now a continuous map . A simplicial map is said to be a simplicial approximation of if and only if each is mapped by onto the support of in . If such an approximation exists, one can construct a homotopy transforming into by defining it on each simplex; there it always exists, because simplices are contractible. The simplicial approximation theorem guarantees for every continuous function the existence of a simplicial approximation at least after refinement of , for instance by replacing by its iterated barycentric subdivision. The theorem plays an important role for certain statements in algebraic topology in order to reduce the behavior of continuous maps on those of simplicial maps, for instance in Lefschetz's fixed-point theorem. Lefschetz's fixed-point theorem The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points. This data is computed as follows: Suppose that and are topological spaces that admit finite triangulations. A continuous map induces homomorphisms between its simplicial homology groups with coefficients in a field . These are linear maps between - vectorspaces, so their trace can be determined and their alternating sum is called the Lefschetz number of . If , this number is the Euler characteristic of . The fixpoint theorem states that whenever , has a fixed-point. In the proof this is first shown only for simplicial maps and then generalized