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The title says it all. I am looking for an explanation or reference for why the homology of the ribbon graph complex computes the cohomology of the mapping class groups of surfaces.

I've seen explanations of this using operads, but my understanding is that the operad viewpoint is more recent and not how the above question was originally understood.

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4 Answers 4

up vote 8 down vote accepted

Chapter 2 of Harer's paper "The cohomology of the moduli space of curves" is good.

The point is that there is the "arc complex", a simplicial complex which gives a suitable triangulation of Teichmüller space which is compatible with the action of the mapping class group. The simplices of the simplicial complex correspond to "arc systems", which are certain collections of curves $C_i$ on an oriented surface $\Sigma$ with boundary $\partial \Sigma$ that begin and end on the boundary, and which decompose the surface into discs.

You can define a ribbon graph to be a graph together with a cyclic ordering of the edges around each vertex. To get a ribbon graph corresponding to an arc system, take the dual graph of the arc system, that is take the graph whose vertices are the components of $\Sigma \setminus (\bigcup_i C_i)$, and the edges are as you'd guess (I can't think of a nice terse way to put this into words, but it's easy to explain with a picture...), and then give the edges around each vertex a cyclic ordering via the orientation of the surface.

Beware that Harer's paper doesn't mention ribbon graphs, nor do some of the other standard references. This had confused me for a while (an embarrassingly long time, in fact) when I was trying to read about this in the literature. But as I said, just know that the ribbon graph picture is just the dual picture to the arc system picture.

P.S. Some more references here.

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The curves should decompose the surface into discs, too. –  Oscar Randal-Williams Jan 13 '11 at 17:22
Right, thanks, I'll add that. –  Kevin H. Lin Jan 13 '11 at 17:33

My favorite way to 'see' the connection between ribbon graphs and mapping class groups is to use the contractibility of the complex of arcs in a surface. Given a surface $S$, the arc complex $\mathcal{A}(S)$ has vertices given by isotopy classes of arcs with endpoints lying on the boundary of the surface, and simplices given by disjoints collections of such arcs. Hatcher has a rather attractive way to construct a contraction of this complex down to a single vertex.

By an easy inductive argument one can deduce from the contractibility of the arc complex that the poset $\mathcal{A}_0(S)$ of simplices for which the arcs cut the surface into pieces that are all discs is also contractible. Such a collection of arcs has a dual ribbon graph.

The mapping class group $MCG(S)$ acts on the poset $\mathcal{A}_0(S)$, and the homotopy quotient is equivalent to the classifying space of the mapping class group since the poset is contractible. On the other hand, one can build a model for the homotopy quotient by taking the category $MCG(S) \int \mathcal{A}_0(S)$ in which objects are objects of $\mathcal{A}_0(S)$ and a morphism $x \to y$ is a mapping class group element $\alpha$ such that $\alpha\cdot x \subset y$ (it is a theorem of Thomason that this models the homotopy quotient). By sending a collection of arcs to its dual ribbon graph one sees that this category is equivalent to the category of ribbon graphs of type $S$.

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In my opinion there are two steps involved in proving the relation. All the proofs I know go as follows: (1) prove that the ribbon graph homology complex computes the cohomology of the geometric realisation of the category $Ribbon$ of ribbon graphs, and (2) show that the geometric realisation of this category is homotopy equivalent to the classifying space of the mapping class group.

For the first step, I like Igusa's proof in chapter one of Graph cohomology and Kontsevich cycles (arXiv:math/0303157v1) The idea is that there is a canonical chain map $C_\ast(|Ribbon|;\mathbb{Q}) \to \mathcal{G}_* \otimes \mathbb{Q}$ up to homotopy, where the first is the cellular homology and the second is the compactly supported graph cohomology complex. This comes from an acyclic carrier over $Ribbon$ known as the forest carrier. The idea is sent a ribbon graph to the chain complex of generated by all ribbon graphs mapping to it (by collapsing subtrees), which is acyclic and augmented over $\mathcal{G}_*$.

For the second step, there are many proofs. I know of at least three distinct ones, but unfortunately only on the details of two of these.

(1) Strebel's proof uses the analytic theory of quadratic differentials. A horizontal trajectory of a non-zero quadratic differential is a curve along which the differential attains real positive values. These trajectories are either closed or non-closed. The closed trajectory come in families and these families decompose the complement of the non-closed trajectories into annuli or punctured disks. In a single annulus or punctured disks all closed trajectories have the same length. A Jenkins-Strebel is then one for which the non-closed trajectories have measure zero. Together with the zeroes these from a ribbon graph. However, one can do even better: it can shown that given a surface of a genus $g$ with $n$ marked points such that $2g+n>0$ and $n$ non-zero real numbers, the surface minus the marked points admits a unique Jenkins-Strebel differential such that the closed curves are of the given lengths. Conversely, given a metric ribbon graph, one can construct an essentially unique Riemann surface. This gives a bijection between $\mathbb{R}_+^n \times \mathcal{M}_{g,n}$ and the space of metric ribbon graphs à la Kontsevich in Intersection theory of the moduli space of curves, which becomes a homeomorphism when picking the correct topologies. It is then not hard to show that the space of metric ribbon graphs is homotopy equivalent to $|Ribbon|$. Kontsevich gives a nice summary of this.

(2) Costello, in A dual point of view on the ribbon graph decomposition of the moduli space of curves (arXiv:math/0601130v1) takes a different route. One proves that the moduli space admits a partial compactification by adding degenerate surfaces which have nodes on the boundary. The inclusion of the moduli space in this partial compactification is a weak equivalence of orbispaces. However, the partial compactification contains a simple subspace $D$ of surfaces whose irreducible components are disks. The inclusion of this orbispace into the partial compactification is again a weak equivalence. Now note that by drawing a vertex for each disk and an edge for each nodal point we obtain a ribbon graph. This gives a orbicell decomposition of $D$ which then proves the theorem.

(3) Penner's proof is mentioned above. I am not too familiar with it, unfortunately.

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The canonical reference is:

@article {MR918455, AUTHOR = {Penner, R. C.}, TITLE = {Perturbative series and the moduli space of {R}iemann surfaces}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {27}, YEAR = {1988}, NUMBER = {1}, PAGES = {35--53}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {32G15 (14H15 57R20)}, MRNUMBER = {918455 (89h:32045)}, MRREVIEWER = {C. Earle}, URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214441648}, }

(look also at other papers of Penner's)

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