Balanced Folding Over a Polygon and Euler Numbers

[摘要] In this paper we introduced a new folding over a polygon we called it balanced folding, then we proved that for a balanced folding of a simply connected surface M there is a subgroup of the group of all homeomorphisms of M that acts 1-transitively on the 2-cells of M. Also we explored the relationship between balanced folding and covering spaces. Finally we obtained a general relation of the Euler number of surfaces which may balance folded over a polygon and we also listed all the possibilities if M is a sphere balanced folded over a triangle and we gave the subgroup mentioned above in each case.

[关键词] Surface;Cellular folding;Singularities;Cayley color graph;1-Transitive;Universal covering;Euler number