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Recall that a ribbon graph is a graph with a cyclic ordering at each vertex and such that each vertex has valence greater than or equal to 3. This cyclic ordering exactly gives one the information to thicken the graph to a surface with boundary and it therefore makes sense to talk about the boundary components of a ribbon graph.

We can make these ribbon graphs into a category $\mathsf{RibbonGr}$ by demanding that morphisms are those surjective morphisms of the underlying graphs which collapse trees while preserving the boundary cycles. This is a theorem of Strebel-Penner-Kontsevich-Igusa (bonus sub-question: is this the correct attribution?) that the geometric realisation of $\mathsf{RibbonGr}$ is homotopy equivalent to the disjoint union of moduli spaces of compact Riemann surfaces with boundary over the set of isomorphism classes of compact Riemann surfaces of genus $g$ and $n$ boundary components except $(g,n) = (0,0), (0,1), (0,2)$.

I like to call this the graph model of the moduli spaces. It can be extended to include labels on the ribbon graphs and Riemann surfaces, e.g. in Godin's work on higher string operations. However, this theorem made me wonder: are there any other generalizations of this theorem? Are there other categories of graphs or some (higher-dimensional) generalization of graphs which geometrically realize to spaces homotopy equivalent to moduli spaces?

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Regarding the bonus question, Costello also has a proof that the ribbon graph category models moduli of Riemann surfaces - see arXiv:math/0402015 and arXiv:math/0601130. –  Jeffrey Giansiracusa Dec 19 '10 at 21:40

3 Answers 3

Yes, graphs come up in various moduli contexts quite a bit! As the previous two answers indicated, $Out(F_n)$ is closely related to graphs. This goes back to the original paper of Vogtmann and Culler where outer space was first introduced. $BOut(F_n)$ can be modelled as the moduli space of metric graphs with first Betti number equal to $n$. If you consider the moduli space of graphs as an orbifold then this has the right integral homotopy type (but if you take the coarse quotient space then it is just a rational classifying space because $Out(F_n)$ acts on outer space with finite stabilisers).

Given a cyclic operad $P$ in the category of topological spaces one can talk about the space of graphs with vertices labelled by $P$. An ordinary graph is the same as a $Comm$-labelled graph since the commutative cyclic operad is just a point in each arity. A ribbon graph is the same as an $Assoc$-labelled graph since in cyclic arity $n$ the associative cyclic operad is the set of cyclic oderings on $n$ letters.

So the Culler-Vogtmann work can be interpreted as saying that $BOut(F_n)$ is the moduli space of rank $n$ $Comm$-graphs. Moduli spaces of Riemann surfaces with marked points are are homotopy equivalent to spaces of $Assoc$-graphs. There are some other results of this type. A Mobius graph is like a ribbon graph but with edges possibly given a half-twist; these are the same as graphs labelled by the cyclic operad $InvAss$ for associative algebras with an involution (perhaps could be called the hermitian associative operad). The spaces of Mobius graphs are homotopy equivalent to the moduli spaces of surfaces with a Klein structure (an unoriented version of complex structure), or equivalently, the classifying spaces of the mapping class groups of unorientable surfaces. Another result of this type is that the space of graphs labelled by the framed little 2-discs cyclic operad is homotopy equivalent to the moduli space of 3-dimensional oriented handlebodies.

To expand on some of Jim's comments a bit further: The connection to graph homology is as follows. First you need to know that there is a duality functor for cyclic operads in chain complexes. Sometimes it is called dg-duality, and sometimes it is called the Bar construction or Koszul duality. (Strictly speaking, the duality is given by a bar construction that produces a cyclic cooperad, followed by applying linear duality to turn in back into a cyclic operad; this construction agrees up to quasi-isomorphism with the koszul duality construction for cyclic operads that are Koszul.) The associative cyclic operad is self-dual. The commutative cyclic operad is dual to the Lie cyclic operad.

The general theorem is that if $P$ is a cyclic operad in topological spaces, then $C_*P$ is a cyclic operad in chain complexes with dual $D(C_*P)$, and $D(C_*P)$-graph homology computes the cohomology of the space of $P$-labelled graphs. This is why $Lie$ graph homology computes the cohomology of $Out(F_n)$ and $Ass$ graph homology computes the cohomology of moduli spaces of Riemann surfaces.

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Outer automorphisms of free groups have a rational classifying space given by metric graphs, called "outer space," first described in a paper by Culler and Vogtmann. The rational cohomology of Out$(F_n)$ is given by a chain complex of "Lie graphs," which are similar to ribbon graphs. A ribbon graph has a cyclic ordering at each vertex, which is really a little element of the associative operad. A Lie graph has its vertices labeled by elements of the Lie operad. See my joint paper with Karen Vogtmann "On a theorem of Kontsevich" for a description of this construction. These two examples are a general case of graph homology, which can be defined for any cyclic operad in the category of vector spaces.

As Domenico mentions, Galatius worked with an integral classifying space to compute the stable homology of Out$(F_n)$, which is a bit more complicated than the rational classifying space. It is defined in terms of graphs embedded in $\mathbb R^\infty$.

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In a sense, each topological space $X$ is a moduli space: it is the moduli space for the functor of continuous maps to $X$. So there is always a tautological moduli problem solved by the geometric realization of any category (and in particular by the geometric realization of a category of graphs).

Therefore I interpret the question in the form "are there examples of moduli problems/classifying spaces a priori unrelated to graph categories which in the end turn out to be homotopy equivalent to the geometric realizations of a category of graphs"? With this interpretation, I think the geometric realization of graph cobordims appearing in Soren Galatius' work on stable homology of automorphism groups of free groups (arXiv:math/0610216) is a nice example.

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