Surface - Construction From Polygons

Construction From Polygons

Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels (A with A, B with B), so that the arrows point in the same direction, yields the indicated surface.

• sphere

• real projective plane

• torus

• Klein bottle

Any fundamental polygon can be written symbolically as follows. Begin at any vertex, and proceed around the perimeter of the polygon in either direction until returning to the starting vertex. During this traversal, record the label on each edge in order, with an exponent of -1 if the edge points opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield

• sphere:
• real projective plane:
• torus:
• Klein bottle: .

Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon (square).

The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem.

Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum.