Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Flexibility of Embeddings of Bouquets of Circles on the Projective Plane and Klein Bottle. | Flexibility of Embeddings of Bouquets of Circles on the Projective Plane and Klein Bottle Yan Yang and Yanpei Liu Department of Mathematics Beijing Jiaotong University Beijing yanyang0206@ ypliu@ Submitted Oct 9 2007 Accepted Nov 15 2007 Published Nov 23 2007 Mathematics Subject Classifications 05C10 05C30 Abstract In this paper we study the flexibility of embeddings of bouquets of circles on the projective plane and the Klein bottle. The numbers of equivalence classes of embeddings of bouquets of circles on these two nonorientable surfaces are obtained in explicit expressions. As their applications the numbers of isomorphism classes of rooted one-vertex maps on these two nonorientable surfaces are deduced. 1 Introduction A surface is a compact 2-dimensional manifold without boundary. It can be represented by a polygon of even edges in the plane whose edges are pairwise identified and directed clockwise or counterclockwise. Such polygonal representations of surfaces can be also written by words. For example the sphere is written as O0 aa where a is paired with a but with the opposite direction of a on the boundary of the polygon. The projective plane the torus and the Klein bottle are represented respectively by aa aba b and p q aabb. In general Op Ị Ị aịbịa ị bị and Nq Ị Ị aịaị denote respectively a surface of i 1 i 1 orientable genus p and a surface of nonorientable genus q. Of course N1 O1 and N2 are respectively the projective plane the torus and the Klein bottle. Every surface is homeomorphic to precisely one of the surface Op p 0 or Nq q 1 9 14 . Suppose A a1a2 at t 1 is a word then A af afaf is called the inverse of A. Let S be the collection of surfaces and let AB be a surface. The following topological transformations and their inverses do not change the orientability and genus of a surface TT 1 Aaa B AB where a 2 AB TT 2 AabBab AcBc where c 2 AB and Supported by NNSF of China under Grant THE ELECTRONIC JOURNAL OF .
