Fundamental polygon

In the mathematics each can in the topological sense closed surface are generated by the sides of a polygon having an even page number in pairs identified . This polygon is called a fundamental polygon .

Fundamental polygon of the sphere : aa ⁻¹

Fundamental polygon of the projective plane : aa

Fundamental polygon of the torus : aba ⁻¹ b ⁻¹

Fundamental polygon of the Klein bottle : aba ⁻¹ b

These polygons can be described by a character string that assigns a symbol to each side. Pages that are identified with each other are given the same symbol. An additional exponent 1 or −1 indicates the orientation of the page.

Canonical form for compact areas (without border)

According to the classification theorem , areas can be divided into three equivalence classes. Each of these classes can be assigned a canonical form of the fundamental polygons:

• a sphere

  ${\displaystyle aa^{-1}}$

• an orientable surface from gender ${\displaystyle n}$

  ${\displaystyle a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}a_{2}b_{2}a_{2}^{-1}b_{2}^{-1}\dots a_{n}b_{n}a_{n}^{-1}b_{n}^{-1}}$

• a non-orientable surface of gender ${\displaystyle n}$

  ${\displaystyle a_{1}a_{1}a_{2}a_{2}\dots a_{n}a_{n}}$

Canonical form for compact surfaces with a border

Areas with a border differ from those without in that they also have a certain number of border components. The canonical form is obtained by adding a corresponding number of boundary components to the fundamental polygons of the unbounded surfaces:

• a sphere with edge components ${\displaystyle k}$ ${\displaystyle B_{1},\dots ,B_{k}}$

  ${\displaystyle aa^{-1}c_{1}B_{1}c_{1}^{-1}\dots c_{k}B_{k}c_{k}^{-1}}$

• an orientable surface of the sex with edge components ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle B_{1},\dots ,B_{k}}$

  ${\displaystyle a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\dots a_{n}b_{n}a_{n}^{-1}b_{n}^{-1}c_{1}B_{1}c_{1}^{-1}\dots c_{k}B_{k}c_{k}^{-1}}$

• a non-orientable surface of the gender with edge components ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle B_{1},\dots ,B_{k}}$

  ${\displaystyle a_{1}a_{1}\dots a_{n}a_{n}c_{1}B_{1}c_{1}^{-1}\dots c_{k}B_{k}c_{k}^{-1}}$

literature

• Hershel M. Farkas and Irwin Kra: Riemann Surfaces. Springer, New York 1980, ISBN 0-387-90465-4 .
• Jurgen Jost: Compact Riemann Surfaces. Springer, New York 2002, ISBN 3-540-43299-X .
• William S. Massey: Algebraic Topology: An Introduction. 1st edition. Springer, Berlin 1967, ISBN 3540902716

Web links

Commons : Fundamental Polygon  - album with pictures, videos and audio files