# Map

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

## Idea[edit | edit source]

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.

## Definitions[edit | edit source]

### Graph embedding[edit | edit source]

A map is a graph embedding of a connected multi-graph (allows multiple edges and self-loops)^{[note 1]} onto a compact connected 2-manifold, such that every connected component of the complement of the embedding is homeomorphic to an open disc.^{[1]}^{[2]}^{[3]} These connected components are the faces of the polyhedron.^{[2]}

### Topological[edit | edit source]

Topologically, a map can be defined as a 2-cell decomposition of a compact connected 2-manifold.^{[2]}

### Graph-encoded map[edit | edit source]

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.^{[4]}

Graph-encoded maps have a bijective correspondence with finite maps defined in terms of graph embeddings, thus these definitions are equivalent.^{[5]}

### Rotation systems[edit | edit source]

Another topology-free definition of maps can be made in terms of permutations acting on a set.^{[6]} A **rotation system** is a triple such that:

- X is a set. Elements of the set are called darts.
- ψ is a permutation on X such that for every dart x and ψ has no fixed points.
- φ is a permutation on X .

From here we can define vertices, edges and faces:

- A vertex is an orbit of a dart under φ . That is for some dart x it is the set of all darts such that for some integer n .
- An edge is an orbit of a dart under ψ . Since ψ is an involution each edge has two darts: x and .
- A face is the orbit a dart traces out by alternating between φ and ψ . Precisely it is the set of darts .

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.^{[7]} 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^{[7]} and can be indicated in diagrams by drawing a bar through the edge.^{[7]} Intuitively when determining the faces of a map crossing barred edges causes the direction of the φ permutation to reverse replacing it with .

## Polyhedral maps[edit | edit source]

A **polyhedral map** is a map whose skeleton has no self-loops or multiple edges, such that the intersection of any two distinct elements is either:^{[8]}

- empty
- a single vertex
- a single edge (including its two vertices)

Polyhedral maps are dyadic and don't permit degeneracies such as dihedra and hosohedra. Polyhedral maps are thus a subset of abstract polyhedra.

## External links[edit | edit source]

- nLab contributors. "Topological map" on nLab.
- nLab contributors. "CW complex" on nLab.
- nLab contributors. "Ribbon graph" on nLab.
- nLab contributors. "Combinatorial map" on nLab.
- Wikipedia contributors. "Regular map (graph theory)".
- Wikipedia contributors. "Graph-encoded map".
- Wikipedia contributors. "Rotation system".
- Wedd, N. Regular Maps

## Notes[edit | edit source]

## References[edit | edit source]

- ↑ Wilson (2012:266)
- ↑
^{2.0}^{2.1}^{2.2}Nedela (2007:6) - ↑ Lins (1982:171)
- ↑ Lins (1982:173)
- ↑ Lins (1982:173-175)
- ↑ Bonnington & Little (1995:24)
- ↑
^{7.0}^{7.1}^{7.2}Bonnington & Little (1995:25) - ↑ Brehm & Schulte (1997:3)

## Bibliography[edit | edit source]

- Bonnington; Little (1995).
*The Foundations of Topological Graph Theory*(1 ed.). Springer. doi:10.1007/978-1-4612-2540-9. ISBN 978-1-4612-7573-2. - Brehm, Ulrich; Schulte, Egon (1997),
*Polyhedral Maps* - Lins, Sóstenes (1982). "Graph-encoded maps".
*Journal of Combinatorial Theory*. Series B.**32**(2): 171–181. doi:10.1016/0095-8956(82)90033-8. MR 0657686. - Nedela, Roman (2007),
*Maps, Hypermaps, and Related Topics*(PDF) - Wilson, Steve (2012). "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators".
*Symmetry*. doi:10.3390/sym4020265.

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