Map

Maps are a concrete definition of a polyhedron based on its topological surface.

Idea
We can think about polyhedra as tilings of a surface. Euclidean tilings are tilings of the plane, and convex polytopes are tilings of a sphere-like surface. We'd like to extend this idea to tile more surfaces than these. Maps are a topological way of extending this concept of tiling a space so that we can tile all sorts of surfaces.

A map is a division of a surface into vertices, edges and faces. Vertices should be single points (0-balls), edges should look like line segments (1-balls) and faces should look like discs (2-balls). We want these divisions to be nicely behaved somehow, so making every point on a surface a vertex isn't a map, decomposing the surface into only edges isn't a map et cetera. We want maps to look somewhat like polyhedra.

Graph embedding
A map is a graph embedding of a connected multi-graph (allows multiple edges and self loops) onto a compact connected 2-manifold, such that every connected component of the compliment of the embedding is homeomorphic to an open disc. These connected components are the faces of the polyhedron.

Topological
Topologically, a map can be defined as a 2-cell decomposition of a compact connected 2-manifold.

Graph-encoded map
The graph-encoded map of a tetrahedron on the left and on the right the same map with its $e$ edges removed to show a graph made only of cycles of size 4.

A map can also be defined without reference to topology at all as a graph-encoded map. A graph-encoded map or gem is a finite properly edge 3-colored graph, with colors $v$, $e$ and $f$ such that the subgraph generated by the edges $v$ and $f$ form cycles of size 4.

Graph-encoded maps have a bijective correspondence with finite maps defined in terms of graph embeddings, thus these definitions are equivalent.