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.

Rotation systems


Another topology-free definition of maps can be made in terms of permutations acting on a set. A rotation system is a triple $$(X,\psi,\phi)$$ such that:
 * $X$ is a set. Elements of the set are called darts.
 * is a permutation on $X$ such that $$\psi(\psi(x)) = x$$ for every dart $x$ and has no fixed points.
 * is a permutation on $X$.

From here we can define vertices, edges and faces:


 * A vertex is an orbit of a dart under . That is for some dart $x$ it is the set of all darts such that $$y = \psi^n(x)$$ for some integer $n$.
 * An edge is an orbit of a dart under . Since is an involution each edge has two darts: $x$ and $$\psi(x)$$.
 * A face is the orbit a dart traces out by alternating between and .  Precisely it is the set of darts $$\left\{\varphi\circ(\psi\circ\varphi)^i(x), (\psi\circ\varphi)^i(x)\mid i\in \mathbb{Z}\right\}$$.

Two elements are incident on another if their intersection is non-empty.

Rotation systems correspond exactly to orientable maps. That is every orientable map under the other definitions is expressible as a rotation system and vice versa. Rotation systems can be extended to make a definition of map that encompasses non-orientable maps by making some edges reverse orientation. Edges that reverse orientation are called barred edges and can be indicated in diagrams by drawing a bar through the edge. Intuitively when determining the faces of a map crossing barred edges causes the direction of the permutation to reverse replacing it with $$\varphi^{-1}$$.