nLab
ribbon graph

Contents

Idea

A ribbon graph (also called fat graph ) is a finite connected graph equipped with a cyclic ordering on the half edges incident to each vertex; it is also required that the valence of each vertex is at least 3.

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 definion of an ordinary graph 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)(V, H, s, i), where

  • VV is a set, called the set of vertices;

  • HH is a set, called the set of half-edges

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

  • i:HHi : 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 ii is the set EE of full edges.

A homomorphism of graphs is a pair of functions V 1V 2V_1 \to V_2 and H 1H 2H_1 \to H_2 commuting with the ii- and ss-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)(V,H,i,s) equipped with a bijection σ:HH\sigma : H \to H whose cycles correspond to the sets s 1(v)s^{-1}(v) of half-edges incident on vertices vVv \in V.

This style of definition apparently goes back to (Igusa 02).

Remark

This means that on all sets s 1(v)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.

Boundary of a fat graph

Definition

Write

:FatGraphGraph \partial : FatGraph \to Graph

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

Γ=( vV hs 1(v){v h}, hH{h 0,h 1},s Γ,i Γ) \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 Γs_{\partial \Gamma} given by

s Γ(h 0)=v h s_{\partial \Gamma}(h_0) = v_h
s Γ(h 1)=σ(v h) s_{\partial \Gamma}(h_1) = \sigma(v_h)

and i Γi_{\partial \Gamma} given by

i Γ(h 0)=i(h) 1 i_{\partial \Gamma}(h_0) = i(h)_1
i Γ(h 1)=i(h) 0. i_{\partial \Gamma}(h_1) = i(h)_0 \,.
Proposition

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

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

Proposition/Definition

There is a natural transformation

c:S c : \partial \to S

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

v hs(h) v_h \mapsto s(h)

and on half-edges

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

Further constructions

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

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

A morphism of graphs Γ 0Γ 1\Gamma_0\to\Gamma_1 is defined as an isomorphism Γ 0/FΓ 1\Gamma_0/F\to\Gamma_1 where FF is a spanning forest of Γ 0\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 Γ/F\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. Γ 0/FΓ 1\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 Γ\Gamma and Γ\Gamma', the set Aut(Γ)Aut(\Gamma) acts freely on Hom(Γ,Γ)Hom(\Gamma,\Gamma') on the right and Aut(Γ)Aut(\Gamma') acts freely on Hom(Γ,Γ)Hom(\Gamma,\Gamma') from the left.

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

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

M g sM^s_g is a mapping class group of a surface of genus gg with ss 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 VV of vertices;

  • a set HH of half-edges;

  • a source map s:HVs : H \to V;

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

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

Geometric realization

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

Proposition

The genus of Σ Γ\Sigma_\Gamma is

g=12(|V||E||bounarycomponents|+2). g = \frac{1}{2} \left( - \vert V\vert - \vert E\vert - \vert bounary components\vert + 2 \right) \,.
Proposition

For every surface SS with boundary, there is a fat graph Γ\Gamma and a homeomorphism

SΣ Γ. S \stackrel{\simeq}{\to} \Sigma_\Gamma \,.

Moduli space of surfaces / classifying spaces of mapping class groups

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

Theorem

There is a weak homotopy equivalence

|FatGraph 3 c| [Σ]BΓ Σ, \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 n1n \geq 1 marked points, except those for which (g,n)=(0,1)(g,n) = (0,1) or (g,n)=(0,2)(g, n) = (0,2).

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

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

Theorem

There is a weak homotopy equivalence

|FatGraph c|BU(1)BU(1) [Σ]BΓ Σ, \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 n1n \geq 1 marked points, except those for which (g,n)=(0,1)(g,n) = (0,1) or (g,n)=(0,2)(g, n) = (0,2). The two extra copies of BU(1)B U(1) corespond to these two exceptional cases.

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

References

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 ¯\overline{\mathbb{Q}}. Mikio Sato: a great Japanese mathematician of the twentieth century. Asian J. Math. 2 (1998), no. 4, 875–919. (pdf)

  • 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, K. Vogtmann, On a theorem of Kontsevich, math.QA/0208169

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

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

A survey is in chapter 3 of

Some discussion is at

Revised on April 6, 2013 22:42:11 by Todd Trimble (67.81.93.26)