The definition of simplicial complex and manifolds allows us to understand two general properties of surfaces, which are Euler characteristic and orientability. Using the Dehn-Sommerville equations written with f- and h-vectors we define the Euler characteristic. We see that this, in addition to orientability, leads us to classification theorem of surfaces. Up to a homeomorphism we get one of the surfaces: a sphere, a connected sum of n tori or a connected sum of n projective planes. We present the surfaces in two different ways: with polygon and with Heffter presentation. The main finding in this thesis is the minimal triangulation of surfaces. We show Heawood conjecture on examples of the sphere, torus, double torus, projective plane and Klein bottle.
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