View source for Map

{{for | a correspondence between two sets | Function}}
'''Maps''' are a concrete definition of a [[polyhedron]] based on its topological surface.

== Idea ==
[[File:Icos-tiling.png|thumb|250px|The [[icosahedron]] as a tiling of the sphere.]]
We can think about polyhedra as tilings of a surface. [[Euclidean tiling]]s are tilings of the plane, and [[convex polytope]]s 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 ==

=== Graph embedding ===
A map is a [[graph embedding]] of a connected multi-graph (allows multiple edges and self-loops)<ref group=note>Some authors require that the graph be [[simple graph|simple]] (i.e. without multiple edges or self-loops) with every vertex having [[degree]] at least 3, other authors refer to these maps as '''singular''' maps. This wiki follows the latter convention, using the term map to refer to a broader class of structures.</ref> onto a compact connected 2-[[manifold]], such that every connected component of the complement of the embedding is homeomorphic to an open disc.<ref>{{harvcoltxt|Wilson|2012|page=266}}</ref><ref name="Nedela6">{{harvcoltxt|Nedela|2007|p=6}}</ref><ref name="Lins171">{{harvcoltxt|Lins|1982|p=171}}</ref>  These connected components are the [[face]]s of the polyhedron.<ref name="Nedela6"/>

=== Topological ===
Topologically, a map can be defined as a 2-cell decomposition of a compact connected 2-manifold.<ref name="Nedela6"/>

=== Graph-encoded map ===
<div class="thumb tright"><div class="thumbinner" style="width:402px;"><div class="thumbimage noresize" style="width:auto;">
[[Image:Tetrahedron flag graph.svg|200px]][[Image:Tetrahedron flag graph edges.svg|200px]]
</div>
<div class="thumbcaption">
The graph-encoded map of a [[tetrahedron]] on the left and on the right the same map with its {{mvar|e}} edges removed to show a graph made only of cycles of size 4.
</div></div></div>
{{See also|Flag graph|Complex}}
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 [[Graph#Properly edge colored graphs|properly edge 3-colored graph]], with colors {{mvar|v}}, {{mvar|e}} and {{mvar|f}} such that the subgraph generated by the edges {{mvar|v}} and {{mvar|f}} form cycles of size 4.<ref name="Lins173">{{harvcoltxt|Lins|1982|p=173}}</ref>

Graph-encoded maps have a [[bijective]] correspondence with finite [[#Graph embedding|maps defined in terms of graph embeddings]], thus these definitions are equivalent.<ref>{{harvcoltxt|Lins|1982|p=173-175}}</ref>

=== Rotation systems ===
[[Image:Rotation System of a Tetrahedron.svg|thumb|300px|The rotation system of a [[tetrahedron]]. Red double-tipped arrows indicate the {{mgreek|phi}} permutation, blue bi-directional arrows indicate the {{mgreek|psi}} permutation.]]
{{main|Rotation system}}
Another topology-free definition of maps can be made in terms of permutations acting on a set.<ref>{{harvcoltxt|Bonnington|Little|1995|p=24}}</ref> A '''rotation system''' is a triple <math>(X,\psi,\phi)</math> such that:
* {{mvar|X}} is a set. Elements of the set are called darts.
* {{mgreek|psi}} is a permutation on {{mvar|X}} such that <math>\psi(\psi(x)) = x</math> for every dart {{mvar|x}} and {{mgreek|psi}} has no fixed points.
* {{mgreek|phi}} is a permutation on {{mvar|X}}.

From here we can define vertices, edges and faces:

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

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.<ref name="FoTGT25">{{harvcoltxt|Bonnington|Little|1995|p=25}}</ref> 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<ref name="FoTGT25" /> and can be indicated in diagrams by drawing a bar through the edge.<ref name="FoTGT25"/> Intuitively when determining the faces of a map crossing barred edges causes the direction of the {{mgreek|phi}} permutation to reverse replacing it with <math>\varphi^{-1}</math>.

== Polyhedral maps ==
A '''polyhedral map''' is a map whose [[skeleton]] has no self-loops or multiple edges, such that the intersection of any two distinct [[element]]s is either:<ref>{{harvcoltxt|Brehm|Schulte|1997|page=3}}</ref>
* 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 polytope|abstract polyhedra]].

== External links ==
* {{nLab|Topological map}}
* {{nLab|override=CW complex}}
* {{nLab|Ribbon graph}}
* {{nLab|Combinatorial map}}
* {{Wikipedia|Regular map (graph theory)}}
* {{Wikipedia|Graph-encoded map}}
* {{Wikipedia|Rotation system}}
* Wedd, N. [https://www.weddslist.com/rmdb/index.php Regular Maps]

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}

== Bibliography ==
* {{ cite book
  | last1 =
    Bonnington
  | last2 =
    Little
  | title =
    The Foundations of Topological Graph Theory
  | publisher =
    Springer
  | year =
    1995
  | doi =
    10.1007/978-1-4612-2540-9
  | isbn =
    978-1-4612-7573-2
  | edition =
    1
  }}
* {{ citation
  | last1 =
    Brehm
  | first1 =
    Ulrich
  | last2 =
    Schulte
  | first2 =
    Egon
  | title =
    Polyhedral Maps
  | year =
    1997
  }}
* {{ cite journal
  | last =
    Lins
  | first =
    Sóstenes
  | issue =
    2
  | journal =
    Journal of Combinatorial Theory
  | mr =
    657686
  | pages =
    171-181
  | series =
    Series B
  | title =
    Graph-encoded maps
  | doi =
    10.1016/0095-8956(82)90033-8
  | volume =
    32
  | year =
    1982
  | doi-access =
    free
  }}
* {{ citation
  | last =
    Nedela
  | first =
    Roman
  | title =
    Maps, Hypermaps, and Related Topics
  | url =
    https://www.savbb.sk/~nedela/CMbook.pdf
  | year =
    2007
  }}
* {{ cite journal
  | last =
    Wilson
  | first =
    Steve
  | url =
    https://www.mdpi.com/2073-8994/4/2/265/htm
  | title =
    Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators
  | doi =
    10.3390/sym4020265
  | date =
    2012
  | journal =
    Symmetry
  | doi-access =
    free
  }}

{{stub}}

{{Polytope-like structures}}

[[Category:Topology]]