Construction from polygons Surface (topology)

any fundamental polygon can written symbolically follows. begin @ vertex, , proceed around perimeter of polygon in either direction until returning starting vertex. during traversal, record label on each edge in order, exponent of -1 if edge points opposite direction of traversal. 4 models above, when traversed clockwise starting @ upper left, yield

sphere:



a
b

b

−
1



a

−
1




{\displaystyle abb^{-1}a^{-1}}


real projective plane:



a
b
a
b


{\displaystyle abab}


torus:



a
b

a

−
1



b

−
1




{\displaystyle aba^{-1}b^{-1}}


klein bottle:



a
b
a

b

−
1




{\displaystyle abab^{-1}}

.

note sphere , projective plane can both realized quotients of 2-gon, while torus , klein bottle require 4-gon (square).

the expression derived fundamental polygon of surface turns out sole relation in presentation of fundamental group of surface polygon edge labels generators. consequence of seifert–van kampen theorem.

gluing edges of polygons special kind of quotient space process. quotient concept can applied in greater generality produce new or alternative constructions of surfaces. example, real projective plane can obtained quotient of sphere identifying pairs of opposite points on sphere. example of quotient connected sum.