Fundamental polygon
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:
In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere.
In the second case of genus one, the surface is conformally equivalent to a torus C/Λ for some lattice Λ in C. The fundamental polygon of Λ is either a period parallelogram or a centrally symmetric polygon, a result first proved by Fedorov in 1891.
In the last case of genus g > 1, the Riemann surface is conformally equivalent to H/Γ where Γ is a Fuchsian group of Möbius transformations. A fundamental domain for Γ is given by a convex polygon for the hyperbolic metric on H. These can be defined by Dirichlet polygons and have an even number of sides. The structure of the fundamental group Γ can be read off from such a polygon. Using the theory of quasiconformal mappings and the Beltrami equation, it can be shown there is a canonical convex Dirichlet polygon with 4g sides, first defined by Fricke, which corresponds to the standard presentation of Γ as the group with 2g generators a1, b1, a2, b2, ..., ag, bg and the single relation [a1,b1][a2,b2] ⋅⋅⋅ [ag,bg] = 1, where [a,b] = a b a−1b−1.
Any Riemannian metric on an oriented closed 2-manifold M defines a complex structure on M, making M a compact Riemann surface. Through the use of fundamental polygons, it follows that two oriented closed 2-manifolds are classified by their genus, that is half the rank of the Abelian group Γ/[Γ,Γ], where Γ = π1(M). Moreover, it also follows from the theory of quasiconformal mappings that two compact Riemann surfaces are diffeomorphic if and only if they are homeomorphic. Consequently, two closed oriented 2-manifolds are homeomorphic if and only if they are diffeomorphic. Such a result can also be proved using the methods of differential topology.
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