Maps as a mathematical main topic arose probably from the four color problem and the more general map coloring problem in the mid of the nineteenth century. Author could not list even main references on them because it is well known for mathematicians and beyond the scope of this lecture notes. Here, author only intends to present a comprehensive theory of combinatorial maps as a rigorous mathematical concept which has been developed only in recent few decades.
The beginning of maps in mathematics was as a topological, or geometric object even with geographical consideration. The first formal definition of a map was done by Heffter in the 19th century. However, it was not paid an attention by mathematicians until 1960 when Edmonds published a note in the AMS Notices with the dual form of Heffter's...