In this chapter we establish some basic tools for graphs embedded on
surfaces. We restrict ourselves to compact 2-dimensional surfaces
without boundary. These are compact connected
Hausdorff topological spaces in which
every point has a neighborhood that is homeomorphic to the plane R^{2}.
In Section 3.1 we present a simple rigorous proof (due to
Thomassen [Th92] of the fact that every surface is homeomorphic to a
triangulated surface. Other available proofs of this result are
complicated and appeal to geometric intuition. Perhaps it is difficult
to follow some of the details but the proof is conceptually
very simple since it merely consists of repeated use of the
Jordan-Schoenflies Theorem. The surface classification theorem, which we
prove next, says that every surface is homeomorphic to a space obtained from
the sphere by adding handles and crosscaps. One of the first complete
proofs of this fundamental result was given by Kerekjarto
[Ke23]. The proof presented in Section 3.1
is very short and follows [Th92]. It differs from other proofs
in the literature in that it uses no topological results, not even the
Jordan Curve Theorem (except for polygonal simple closed curves in the
plane). In particular, it does not use Euler's formula
(which includes the Jordan Curve Theorem). Instead, Euler's formula
is a by-product of the proof.
After obtaining the surface classification theorem, it is shown that every
orientable graph embedding, whose faces are all homeomorphic to
an open disc in the plane, can be described
combinatorially, up to homeomorphism, by the local rotations at each vertex.
A similar description exists also for embeddings of graphs into
nonorientable surfaces. Again, this is proved rigorously without appealing
to geometric intuition.

Inspired by the possibility of describing embeddings of graphs
combinatorially, we **define** in Chapter 4
an embedding as a collection of local rotations in the orientable case
and, more generally, an embedding scheme in the nonorientable case.
In the remaining part of the book we treat embeddings in this purely
combinatorial framework.