A ribbon graph (also called fat graph ) is a (typically finite and connected) graph equipped with a cyclic ordering on the half edges incident to each vertex.
To each ribbon graph, one can associate an oriented surface with boundary by replacing edges by thin oriented rectangles (ribbons) and vertices by disks, and pasting rectangles to disks according to the chosen cyclic orders at the vertices. (For some illustrations of this, see Mulase-Penkava.)
The following definition of an ordinary graph (allowing loops and multiple edges) is not the standard one used for ordinary graphs, but is easily seen to be equivalent to it, while being the right starting point for describing ribbon graphs.
A graph is a quadruple , where
is a set, called the set of vertices;
is a set, called the set of half-edges
is a function, thought of as sending each half-edge to the vertex that it is incident on;
The set of cycles of is the set of full edges.
An edge is a loop if its constituent half-edges are incident on the same vertex.
This means that on all sets of jointly incident half-edges there is induced a cyclic ordering. When drawing ribbon graphs on paper this cyclic ordering is identified with the ordering induced from the standard orientation of the plane.
Note that this definition (which appears in Igusa 02) is essentially equivalent to that of an oriented combinatorial map, except that the usual definition of combinatorial map does not make the set and the function explicit, instead identifying vertices with the cycles of . Under this style of definition, the extra condition (often assumed) that the underlying graph be connected corresponds to the condition that the group generated by the permutations and acts transitively on the set of half-edges .
for the functor – called the boundary graph functor – that sends a fat graph to the graph
with given by
and given by
Write for the evident forgetful functor sending fat graphs to their underlying graph.
There is a natural transformation
from the boundary graph functor to the forgetful functor – called the collapse map, whose components morphism on a fat graph sends on vertices
and on half-edges
Given any edge in a graph , we can collapse it to a new graph by merging the two vertices containing the edge and removing both half edges.
A forest (disjoint union of trees) spans a graph if it contains all of its vertices. One can collapse to by collapsing each tree to a separate vertex. Then the edges in are the edges of which do not lie in any tree in ; the vertices of correspond to the sets of leaves of the component trees of .
A morphism of graphs is defined as an isomorphism where is a spanning forest of . Morphisms of such graphs are both epi and mono.
For a ribbon graph, the set of leaves of any tree inherits a cyclic order. Thus has a canonical structure a ribbon graph. This makes the following definition possible: A morphism of ribbon graphs is the morphism of underlying graphs which respects the ordering on the vertices; i.e. is an isomorphism of ribbon graphs. Every endomorphism of a ribbon graph is an automorphism and for any two ribbon graphs and , the set acts freely on on the right and acts freely on from the left.
Let denote the category of ribbon graphs and ribbon automorphisms. Strebel proved that the geometric realization of the category of ribbon graphs decomposes into a direct sum of classifying spaces of all mapping class groups
is a mapping class group of a surface of genus with punctures. Alternatively, one can instead of the geometric realization take a moduli space of ribbon graphs with metric. A metric on a ribbon graph is a positive real valued function on the set of edges.
For the following we represent a fat graph by
a set of vertices;
a set of half-edges;
a source map ;
an involution which sends each half-edge to its partner half-edge;
an permutation which gives the cyclic order on the edges.
For a fat graph, write for the surface that it defines.
The genus of is
For every surface with boundary, there is a fat graph and a homeomorphism
Write for the full subcategory of fat graphs on the connected fat graphs for which every vertex has valence at least 3. Write Top for the geometric realization of this category. For a surface, write for its mapping class group and for the corresponding classifying space.
There is a weak homotopy equivalence
where on the right thr coproduct ranges over isomorphism classes of orientable, closed 2-dimensional manifolds with marked points, except those for which or .
The restriction to valence can be dropped:
There is a weak homotopy equivalence
where again on the right the coproduct ranges over isomorphism classes of orientable, closed 2-dimensional manifolds with marked points, except those for which or . The two extra copies of corespond to these two exceptional cases.
Original references include
Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. (1992), no. 147, 1-23.
Kiyoshi Igusa, Higher Franz Reidemeister torsion , IP Studies in Advanced Mathematics, American Mathematical Society, 2002.
K. Strebel, Quadratic Differentials, Springer, Berlin, 1984, MR86a:30072
R. C. Penner, The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys. 113 (2) (1987) 299–339. MR89h:32044
John Harer, The cohomology of the moduli space of curves, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.
M. Mulase, M. Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over . Mikio Sato: a great Japanese mathematician of the twentieth century. Asian J. Math. 2 (1998), no. 4, 875–919. (pdf)
J. Conant, K. Vogtmann, On a theorem of Kontsevich, math.QA/0208169
A survey is in chapter 3 of
Some discussion is at