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I can think of a couple of other reasons. The first is algorithms. Many algorithmic problems become easier if we are told that the input graph is embeddable on a surface. As a trivial example: the problem of deciding if a graph is 4-colourable is NP-hard in general, but pretty damn easy when restricted to the class of planar graphs. So, that begs the question: how do we specify an embedding of a graph in a finite way? The geometric definition that you propose is an infinite beast which we can't really feed a computer. On the other hand, it is easy to tell a computer what a rotation system is.

Secondly, we don't really lose anything with this approach. That is, most of the questions about graphs that we care about don't really depend on how the edges of the graph are drawn on the surface. One can prove that up to mucking around on the surface, rotation systems really do gives us all embeddings of graphs on a surface.

Finally, while it is perfectly reasonable to define an embedding of a graph geometrically, you actually haven't said how to define the dual graph given a geometric embeddinggeometrically. For example, how do you propose to geometrically define the dual of a tree embedded on say a torus?

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I can think of a couple of reasons. The first is algorithms. Many algorithmic problems become easier if we are told that the input graph is embeddable on a surface. As a trivial example: the problem of deciding if a graph is 4-colourable is NP-hard in general, but pretty damn easy when restricted to the class of planar graphs. So, that begs the question: how do we specify an embedding of a graph in a finite way? The geometric definition that you propose is an infinite beast which we can't really feed a computer. On the other hand, it is easy to tell a computer what a rotation system is.

Secondly, we don't really lose anything with this approach. That is, most of the questions about graphs that we care about don't really depend on how the edges of the graph are drawn on the surface. One can prove that up to mucking around on the surface, rotation systems really do gives us all embeddings of graphs on a surface.

Finally, while it is perfectly reasonable to define an embedding of a graph geometrically, you actually haven't said how to define the dual graph given a geometric embedding. For example, how do you propose to geometrically define the dual of a tree embedded on say a torus?