nLab ribbon graph

Idea

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.)

Definition

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.

Definition

A graph is a quadruple $(V, H, s, i)$, where

• $V$ is a set, called the set of vertices;

• $H$ is a set, called the set of half-edges

• $s : H \to V$ is a function, thought of as sending each half-edge to the vertex that it is incident on;

• $i : H \to H$ is a involutive function without fixed points, thought of as sending each half-edge to its other half.

The set of cycles of $i$ is the set $E$ of full edges.

A homomorphism of graphs is a pair of functions $V_1 \to V_2$ and $H_1 \to H_2$ commuting with the $i$- and $s$-maps.

An edge is a loop if its constituent half-edges are incident on the same vertex.

Definition

A fat graph or ribbon graph is a graph $(V,H,i,s)$ equipped with a bijection $\sigma : H \to H$ whose cycles correspond to the sets $s^{-1}(v)$ of half-edges incident on vertices $v \in V$.

This means that on all sets $s^{-1}(v)$ 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 $V$ and the function $s : H \to V$ explicit, instead identifying vertices with the cycles of $\sigma$. 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 $i$ and $\sigma$ acts transitively on the set of half-edges $H$.

Boundary of a fat graph

Definition

Write

$$\partial : FatGraph \to Graph$$

for the functor – called the boundary graph functor – that sends a fat graph $\Gamma = (V, H, s, i, \sigma)$ to the graph

$$\partial \Gamma = \left( \coprod_{v \in V} \coprod_{h \in s^{-1}(v)} \{v_h\},\, \coprod_{h \in H} \{h_0, h_1\},\, s_{\partial_\Gamma},\, i_{\partial_\Gamma} \right)$$

with $s_{\partial \Gamma}$ given by

$$s_{\partial \Gamma}(h_0) = v_h$$

$$s_{\partial \Gamma}(h_1) = \sigma(v_h)$$

and $i_{\partial \Gamma}$ given by

$$i_{\partial \Gamma}(h_0) = i(h)_1$$

$$i_{\partial \Gamma}(h_1) = i(h)_0 \,.$$

Proposition

The set of boundary components of $\Sigma_\Gamma$ is naturally identified with the cycles of $\sigma \circ i : H \to H$, or equivalently with the cycles of $i \circ \sigma$.

Write $S : FatGraph \to Graph$ for the evident forgetful functor sending fat graphs to their underlying graph.

Proposition/Definition

There is a natural transformation

$$c : \partial \to S$$

from the boundary graph functor to the forgetful functor – called the collapse map, whose components morphism on a fat graph $\Gamma$ sends on vertices

$$v_h \mapsto s(h)$$

and on half-edges

$$h_{0,1} \mapsto h \,.$$

Further constructions

Given any edge $e$ in a graph $\Gamma$, we can collapse it to a new graph $\Gamma/e$ by merging the two vertices containing the edge and removing both half edges.

A forest $F$ (disjoint union of trees) spans a graph $\Gamma$ if it contains all of its vertices. One can collapse $\Gamma$ to $\Gamma/F$ by collapsing each tree to a separate vertex. Then the edges in $\Gamma/F$ are the edges of $\Gamma$ which do not lie in any tree in $F$; the vertices of $\Gamma/F$ correspond to the sets of leaves of the component trees of $F$.

A morphism of graphs $\Gamma_0\to\Gamma_1$ is defined as an isomorphism $\Gamma_0/F\to\Gamma_1$ where $F$ is a spanning forest of $\Gamma_0$. 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 $\Gamma/F$ 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. $\Gamma_0/F\to\Gamma_1$ is an isomorphism of ribbon graphs. Every endomorphism of a ribbon graph is an automorphism and for any two ribbon graphs $\Gamma$ and $\Gamma'$, the set $Aut(\Gamma)$ acts freely on $Hom(\Gamma,\Gamma')$ on the right and $Aut(\Gamma')$ acts freely on $Hom(\Gamma,\Gamma')$ from the left.

Let $\mathcal{Fat}$ denote the category of ribbon graphs and ribbon automorphisms. Strebel proved that the geometric realization $|\mathcal{Fat}|$ of the category of ribbon graphs decomposes into a direct sum of classifying spaces of all mapping class groups $M^s_g$

$$|\mathcal{Fat}| = \coprod_{g,s} BM^s_g$$

$M^s_g$ is a mapping class group of a surface of genus $g$ with $s$ 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.

Properties

For the following we represent a fat graph by

• a set $V$ of vertices;

• a set $H$ of half-edges;

• a source map $s : H \to V$;

• an involution $i : H \to H$ which sends each half-edge to its partner half-edge;

• an permutation $\sigma : H \to H$ which gives the cyclic order on the edges.

Geometric realization

For $\Gamma$ a fat graph, write $\Sigma_\Gamma$ for the surface that it defines.

Proposition

The genus of $\Sigma_\Gamma$ is

$$g = \frac{1}{2} \left( 2 - \vert V\vert + \vert E\vert - \vert boundary components\vert \right) \,.$$

Proposition

For every surface $S$ with boundary, there is a fat graph $\Gamma$ and a homeomorphism

$$S \stackrel{\simeq}{\to} \Sigma_\Gamma \,.$$

Moduli space of surfaces / classifying spaces of mapping class groups

Write $FatGraph^c_3$ for the full subcategory of fat graphs on the connected fat graphs for which every vertex has valence at least 3. Write $\vert FatGraph^c_3 \vert \in$ Top for the geometric realization of this category. For $\Sigma$ a surface, write $\Gamma_Sigma$ for its mapping class group and $B \Gamma_\Sigma$ for the corresponding classifying space.

Theorem

There is a weak homotopy equivalence

$$\vert FatGraph^c_3 \vert \simeq \coprod_{[\Sigma]} B \Gamma_\Sigma \,,$$

where on the right thr coproduct ranges over isomorphism classes of orientable, closed 2-dimensional manifolds with $n \geq 1$ marked points, except those for which $(g,n) = (0,1)$ or $(g, n) = (0,2)$.

In various guises this theorem has been proven by Costello, Kontsevich92, Igusa02, Penner, Strebel.

The restriction to valence $\geq 3$ can be dropped:

Theorem

There is a weak homotopy equivalence

$$\vert FatGraph^c \vert \simeq B U(1) \coprod B U(1) \coprod_{[\Sigma]} B \Gamma_\Sigma \,,$$

where again on the right the coproduct ranges over isomorphism classes of orientable, closed 2-dimensional manifolds with $n \geq 1$ marked points, except those for which $(g,n) = (0,1)$ or $(g, n) = (0,2)$. The two extra copies of $B U(1)$ corespond to these two exceptional cases.

This has been shown in (Godin). One of the $B U(1)$-summands is also produced in (Igusa02). A detailed complete proof appears as (Kupers, theorem 3.59).

Related concepts

• stringy weight system

• graph

• directed graph

  • directed n-graph

• ribbon graph

• topological map

• combinatorial map

• ribbon category

References

Survey in the context of string topology:

• Sander Kupers, Chapter 3 of: String topology operations MSc thesis (2011) (dspace:1874/206062)

Survey in the context of weight systems on chord diagrams/Jacobi diagrams:

• Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 7 of: Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 (arxiv:1103.5628, doi:10.1017/CBO9781139107846)

Original articles:

• 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.

• Kiyoshi Igusa, Graph cohomology and Kontsevich cycles, Topology 43 (2004), n. 6, p. 1469-1510, MR2005d:57028, doi

• 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, 138–221 (1988)

• M. Mulase, M. Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over $\overline{\mathbb{Q}}$. Mikio Sato: a great Japanese mathematician of the twentieth century. Asian J. Math. 2 (1998), no. 4, 875–919. (math-ph/9811024)

• Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics 120, Birkhäuser (1994), 97–121, pdf

• Ib Madsen, Michael Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941, MR2009b:14051, doi

• J. Conant, Karen Vogtmann?, On a theorem of Kontsevich, math.QA/0208169

• Gabriele Mondello, Riemann surfaces, ribbon graphs and combinatorial classes, pdf arxiv:0705.1792

• Veronique Godin, Higher string topology operations (2007)(arXiv:0711.4859)

• Kevin Costello. A dual version of the ribbon graph decomposition of moduli space, Geometry & Topology 11 (2007) 1637–1652 doi

Further discussion

• MathOverflow: why does ribbon graph cohomology compute cohomology of mapping class group