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.Although the book is mainly for researchers in mathematics， theoretical physics， chemistry， biology and some others related， the basic part in each chapter can also be chosen for graduates and college teachers as references.
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