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 exha

Preface
Chapter Ⅰ Preliminaries
Ⅰ.1 Sets and mappings
Ⅰ.2 Partitions and permutations
Ⅰ.3 Group actions
Ⅰ.4 Networks
Ⅰ.5 Notes
Chapter Ⅱ Surfaces
Ⅱ.1 Polyhedra
Ⅱ.2 Elementary equⅣalence
Ⅱ.3 Polyhegons
Ⅱ.4 Orientability
Ⅱ.5 Classification
Ⅱ.6 Notes
Chapter Ⅲ Embeddings of Graphs
Ⅲ.1 Geometric consideration
Ⅲ.2 Surface closed curve axiom
Ⅲ.3 Distinction
Ⅲ.4 Joint tree model
Ⅲ.5 Combinatorial properties
Ⅲ.6 Notes
Chapter Ⅳ Mathematical Maps
Ⅳ.1 Basic permutations
Ⅳ.2 Conjugate axiom
Ⅳ.3 TransitⅣity
Ⅳ.4 Included angles
Ⅳ.5 Notes
Chapter Ⅴ Duality on Surfaces
Ⅴ.1 Dual partition of edges
Ⅴ.5 Notes
Chapter Ⅵ Invariants on Basic Class
Ⅵ.1 Orientability
Ⅵ.2 Euler characteristic
Ⅵ.3 Basic equⅣalence
Ⅵ.4 Orientable maps
Ⅵ.5 Nonorientable maps
Ⅵ.6 Notes
Chapter Ⅶ Asymmetrization
Ⅶ.1 Isomorphisms
Ⅶ.2 Recognition
Ⅶ.3 Upper bound of group order
Ⅶ.4 Determination of the group
Ⅶ.5 Rootings
Ⅶ.6 Notes
Chapter Ⅷ Asymmetrized Census
Ⅷ.1 Orientable equation
Ⅷ.2 Planar maps...
Ⅷ.3 Nonorientable equation
Ⅷ.4 Gross equation
Ⅷ.5 The number of maps
Ⅷ.6 Notes
Chapter Ⅸ Petal Bundles
Ⅸ.1 Orientable petal bundles
Ⅸ.2 Planar pedal bundles
Ⅸ.3 Nonorientable pedal bundles
Ⅸ.4 The number of pedal bundles
Ⅸ.5 Notes
Chapter Ⅹ Super Maps of Genus Zero
Chapter Ⅺ Symmetric Census
Chapter Ⅻ Cycle Oriented Maps
Chapter ⅫⅠ Census by Genus
Chapter ⅩⅣ Classic Applications
Appendix Ⅰ Embeddings and maps of Small Size Distributed by Genus
Appendix Ⅱ Orientable Forms of Surfaces and Their Nonorientable Genus Polynomials
Bibliography
Subject Index
Author Index