In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only information about 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:
the Riemann sphere, the complex plane, the unit disk D or equivalently the upper half-plane H. 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, if assumed convex, may be taken to be either a period parallelogram or a centrally symmetric hexagon, 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 fundamental 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.
Fundamental polygons in genus one
Parallelograms and centrally symmetric hexagons In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R2, it can be assumed if necessary that Λ = Z2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z2 = Z + Z ω, with Im ω > 0.) A fundamental domain is given by the parallelogram s x + t y for 0 < s , t < 1 where x and y are generators of Λ. If C is the interior of a fundamental convex polygon, then the translates C + x cover R2 as x runs over Λ. It follows that the boundary points of C are formed of intersections C ∩ (C + x). These are compact convex sets in ∂C and thus either vertices of C or sides of C. It follows that every closed side of C can be written this way. Translating by −x it follows that C ∩ (C − x) is also a side of C. Thus sides of C occur in parallel pairs of equal length. The end points of two such parallel segments of equal length can be joined so that they intersect and the intersection occurs at the midpoints of the line segments joining the endpoints. It follows that the intersections of al such segments occur at the same point. Translating that point to the origin, it follows that the polygon is centrally symmetric; that is, if a point z is in the polygon, so too is −z. It is easy to see translates of a centrally symmetric convex hexagon tessellate the plane. If A is a point of the hexagon, then the lattice is generated by the displacement vectors AB and AC where B and C are the two vertices which are not neighbours of A and not opposite A. Indeed, the second picture shows how the hexagon is equivalent to the parallelogram obtained by displacing the two triangles chopped off by the segments AB and AC. Equally well the first picture shows another way of matching a tiling by parallelograms with the hexagonal tiling. If the centre of the hexagon is 0 and the vertices in order are a, b, c, −a, −b and −c, then Λ is the Abelian group with generators a + b and b + c.
Examples of fundamental polygons generated by parallelograms Exactly four topologies can be created by identifying the sides of a rhombus in different ways. They are given below as directional edges A and B on a square, either as AABB or ABAB sequences.
Fedorov's theorem Fedorov's theorem, established by the Russian crystallographer Evgraf Fedorov in 1891, asserts that parallelograms and centrally symmetric hexagons are the only convex polygons that are fundamental domains. There are several proofs of this, some of the more recent ones related to results in convexity theory, the geometry of numbers and circle packing, such as the Brunn–Minkowski inequality. Two elementary proofs due to H. S. M. Coxeter and Voronoi will be presented here. Coxeter's proof proceeds by assuming that there is a centrally symmetric convex polygon C with 2m sides. Then a large closed parallelogram formed from N2 fundamental parallelograms is tiled by translations of C which go beyond the edges of the large parallelogram. This induces a tiling on the torus C/NΛ. Let v, e and f be the number of vertices, edges and faces in this tiling (taking into account identifications in the quotient space). Then, because the Euler–Poincaré characteristic of a torus is zero,
v − e + f = 0. {\displaystyle v-e+f=0.}
On the other hand, since each vertex is on at least 3 different edges and every edge is between two vertices,
3 v ≤ 2 e . {\displaystyle 3v\leq 2e.}
Moreover, since every edge is on exactly two faces,
2 e = 2 m f . {\displaystyle 2e=2mf.}
Hence
m f = e ≤ 3 ( e − v ) = 3 f . {\displaystyle mf=e\leq 3(e-v)=3f.}
so that
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