通过对于图在曲面上嵌入的组合结构分析，引出一个公理系统，从而导出Tutte关于地图概念的数学意义。通过在这个公理系统下的严格推导，给出数学地图的拓扑分类，进而在每一个拓扑类中继续根据对称性分为同构类。通过构造原始迹码，或对偶迹码，求得同构类的多项式型的完全不变量。通过不对称化导出Tutte的定根方式，引进根地图的同构类，并证明每一类只有一个根地图。先确定每类地图中含根地图的数目。然后，通过恢复对称性，确定无根地图的数目。 This monograph is for a unified theory of surfaces, embeddings and maps all considered as polyhedra via the joint tree modal which was initiated from the author's articles in the seventies of last century and has been basically developed in recent decades. Complete invariants for each classification are topologically, combinatorially or isomorphically extracted. A number of counting polynomials including handle and crosscap polynomials are presented. In particular, an appendix serves as the exhaustive counting super maps (rooted and nonrooted) including these polynomials with under graphs of small size for the reader's digests.