Results about the Johnson cokernel

Recall that  $\mathfrak{j}(g)$ is the associated graded $\mathbb Q$-Lie algebra for the Johnson filtration of the mapping class group. The Johnson homomorphism embeds this in a Lie algebra of symplectic derivations:$$\tau \colon \mathfrak{j}(g)\to\operatorname{Der}_\omega(\mathsf{L}(H)),$$  where $H=H_1(\Sigma_{g,1};\mathbb Q)$. So in order to understand $\mathfrak{j}(g)$, we look at the essentially equivalent problem of understanding the cokernel $\mathsf{C}(H)$.  (Essentially equivalent because the target $\operatorname{Der}_\omega(\mathsf{L}(H))$ is fairly well understood.)

Here I want to collect some known results about $\operatorname{Sp}(H)$ representations $[\lambda]_{\operatorname{Sp}}$ appearing in $\mathsf{C}(H)$. I will always be working stably, in the sense that these results are true if the genus $g$ is large enough. The symplectic derivation Lie algebra is graded by degree, inducing a grading on $\mathsf{C}(H)=\oplus_{k\geq 1}\mathsf{C}_k(H)$.

In order to give the reader an idea of the complexity of $\mathsf{C}(H)$ we present a table of its decomposition in low degrees: $\newcommand{\SP}{\operatorname{Sp}}$.

•  $\mathsf C_1=\mathsf C_2=0$
• $\mathsf C_3= [3]_{\SP}$
• $\mathsf C_4=[21^2]_{\SP}\oplus[2]_{\SP}$
• $\mathsf C_5=[5]_{\SP}\oplus[32]_{\SP}\oplus[2^21]_{\SP}\oplus[1^5]_{\SP}\oplus 2[21]_{\SP}\oplus2[1^3]_{\SP}\oplus 2[1]_{\SP}$
• $\mathsf C_6=2[41^2]_{\SP} \oplus [3^2]_{\SP} \oplus [321]_{\SP} \oplus [31^3]_{\SP} \oplus [2^21^2]_{\SP} \oplus2[4]_{\SP} \oplus 3[31]_{\SP} \oplus 3[2^2]_{\SP} \oplus 3[21^2]_{\SP} \oplus 2[1^4]_{\SP} \oplus [2]_{\SP} \oplus 5[1^2]_{\SP} \oplus 3[0]_{\SP}$

Early computations were done by Asada, Nakamura, Hain and Morita. The complete table was computed by Morita-Sakasai-Suzuki.

Some of the representations appearing here are parts of known families. Here is a summary of some known constructions.

•  The Galois obstruction, which is an embedding of the Grothendieck-Teichmüller Lie algebra  $\mathfrak{grt}_1\hookrightarrow \mathsf{C}(H)$, appears as $[0]_{\operatorname{Sp}}$ representations, as mentioned in the last post. I believe the map doubles the degree, so that the first nontrivial example $\sigma_3$ appears in $\mathsf{C}_6$.
•  Morita showed early on that $\newcommand{\ext}{\bigwedge\nolimits} \ext^{2k+1}H=[2k+1]_{\operatorname{Sp}}$ appears in $\mathsf{C}_{2k+1}$, using a "trace map," for $k\geq 1$.
• Recently Enomoto and Satoh used a trace map defined by Satoh to detect $[1^{4k+1}]_{\operatorname{Sp}}$ in degree $2k+1$ for $k\geq 1$. More recently, they have used their trace to detect even more classes.
• I showed that the Enomoto-Satoh trace will detect $[H^{\langle n \rangle}]_{D_{2n}}$. Here $H^{\langle n\rangle}\subset H^{\otimes n}$ is the intersection of the kernels of all pairwise contractions $H\otimes H\to \mathbb R$. The dihedral group $D_{2n}$ acts on $H^{\otimes n}$ via $h_1\otimes\cdots\otimes h_n\mapsto h_n\otimes h_1\otimes \cdots\otimes h_{n-1}$ and $h_1\otimes\cdots h_n\mapsto (-1)^{n+1}h_n\otimes\cdots\otimes h_1$, and we take the coinvariants with respect to that action. Calculating $[H^{\langle n \rangle}]_{D_{2n}}$ is purely representation-theoretic and contains Morita's and Enomoto-Satoh's classes from the previous bullet points. In my paper, I did some example representation theory calculations to show that $[H^{\langle n \rangle}]_{D_{2n}}$ is quite large.
• Jointly with Kassabov and Vogtmann, we showed that representations $[2k,2\ell]\otimes\mathcal S_{2k-2\ell+2}$ and $[2k+1,2\ell+1]\otimes \mathcal{M}_{2k-2\ell+2}$ appear in the cokernel in degrees $2k+2\ell+2$ and $2k+2\ell+4$ respectively, where $k>\ell\geq 0$. Here $\mathcal M_w$ and $\mathcal S_w$ refer to the space of modular forms and the space of cusp forms of weight $w$.
• In a forthcoming paper, Kassabov and I show that the $\operatorname{GL}(H)$ decomposition of $\newcommand{\Out}{\operatorname{Out}} H^{2n-3}(\Out(F_n);\overline{T(H)^{\otimes n }})$ appears as an $\operatorname{SP}(H)$ decomposition in the cokernel, for $n\geq 2$, which leads to lots of new representations. Here $T(H)$ is the tensor algebra on $H$. At first sight, it's not obvious that the group $\operatorname{Aut}(F_n)$ acts on $T(H)^{\otimes n}$, but it in fact comes from an action on $\mathcal{H}^{\otimes n}$ for any cocommutative Hopf algebra $\mathcal H$. $\overline{\mathcal H^{\otimes n}}$ is an appropriate quotient on which the action of $\Out(F_n)$ is well-defined. For example, one can detect the family $$[2k,2,1]_{\SP}\otimes \mathcal M_{2k+2}\subset \mathsf C_{2k+5},$$ in this way.

One of my contributions to this story was to define a generalized trace that simultaneously generalized the Morita trace, the Conant-Kassabov-Vogtmann trace, and the Satoh trace. Morita's trace can be thought of graphically as adding a single external edge to a tree connecting two univalent vertices in all possible ways. It makes good sense to call this a trace from the diagrammatic perspective. For example, in Penrose notation for tensors, this is how you define the trace of a matrix, by adding an edge plugging the input into the output. The Satoh trace can be similarly interpreted as adding an edge in all possible ways, but with a different target. Kassabov, Vogtmann and myself introduced a generalization of Morita's map, which we also called a trace, and I now think that may have been a mistake to call it that. It is actually defined as an exponential of Morita's trace and involves summing over adding several extra edges in all possible (unordered ways). Since there is a formula $\det\exp(A)=\exp(\operatorname{tr}(A))$ in matrix algebra, we probably should have called this map the "determinant!" The new trace also sums over adding multiple edges, but the target is defined differently than that for the Conant-Kassabov-Vogtmann trace. In my next post I will go into some more detail about all of these different trace constructions.
 
Right now I'm waiting in the Knoxville airport to head off to Bonn. I'm giving a talk on this stuff on Monday.

The Johnson Homomorphism

The work I've done which has the most direct connection to $\mathfrak{grt}_1$ is on the Johnson homomorphism, so I'd like to explain what that is. We'll consider the mapping class group of a genus $g$ surface with $1$ boundary component $\Sigma_{g,1}$, although the number of  boundary components will not really matter. We will also be taking a limit as $g$ approaches $\infty$ in the end. We denote the mapping class group $\operatorname{Mod}(g,1)$, following the conventions of some authors in thinking of this as the "modular group." Let $x_1,y_1,\ldots,x_g,y_g$ be dual pairs of curves on $\Sigma_{g,1}$ which freely generate the fundamental group $\pi=\pi_1(\Sigma_{g,1})$. Note that any mapping class preserves the boundary, and hence fixes $w_g=\prod_{i=1}^g[x_i,y_i]$. A celebrated theorem of Nielsen states that this is the only condition:

Theorem (Nielsen): The natural map $$\operatorname{Mod}(g,1)\to \{\varphi\in \operatorname{Aut}(\pi)\,|\, \varphi(w_g)=w_g\}$$ is an isomorphism.

One might naively expect that since $\operatorname{Mod}(g,s)$ can be thought of as a subgroup of the automorphism group of a free group, that it would be a fairly straightforward group to analyze, or at least that all problems could be turned into problems about $\operatorname{Aut}(F_g.)$ One problem with this expectation is that $\operatorname{Aut}(F_g)$ is already quite complicated! Furthermore, subgroups do not usually inherit nice properties from their overgroup. Finally, throwing away the geometry throws away many useful tools. (Indeed geometry will end up playing a big role in what follows, but it will be mostly hidden.)

Anyway, the Nielsen isomorphism is instrumental in defining the so-called Johnson filtration of the mapping class group. Given a group $G$, let $G_n$ denote the $n$th term in the lower central series: $G_1=G, G_2=[G,G], G_3=[G,[G,G]],\cdots $ This notation is useful, although it can cause confusion when the group in question is called $\pi$, as it is here. These are not homotopy groups. Define $$\mathbb J_n(g)=\ker(\operatorname{Mod(g,1)} \to \operatorname{Aut}(\pi/\pi_{n+1})),$$ giving rise to the Johnson filtration $$\operatorname{Mod}(g,1)=\mathbb J_0(g)\supset \mathbb J_1(g)\supset \mathbb J_2(g)\supset\cdots.$$ $\mathbb J_1(g)$ is the Torelli group, consisting of those mapping classes that act trivially on homology. The groups $\mathbb J_k(g)$ can be thought of as "higher order" or "deeper" versions of the Torelli group.

Given a filtration, it is natural to consider the associated graded object, $\mathfrak{j}(g)=\oplus \mathfrak j_k(g).$ The group commutator induces a Lie bracket on $\mathfrak{j}(g)$, and after tensoring with a field becomes a Lie algebra. We will consider our field to be $\mathbb Q$. A natural question arises:

Problem: Determine the structure of the Lie algebra $\mathfrak j(g)$.

The first homology, $H$, of $\Sigma_{g,1}$ is isomorphic to $\operatorname{Sp}(H)$, and $\mathfrak j(g)$ turns out to be a $\operatorname{Sp}(H)$ module. Much of the work that has been done has been concentrated on finding a decomposition of $\mathfrak j(g)$ into irreducible symplectic representations. Often one considers the limit as $g$ approaches infinity, looking at the stable decomposition, which is technically easier.

Now consider $\mathsf{L}(H)$, the free Lie algebra on $H$, and the Lie algebra, $\mathsf{Der}(\mathsf{L}(H))$ of derivations of $\mathsf{L}(H)$. There is fairly obvious map $$\tau\colon \mathfrak{j}(g)\to \mathsf{Der}(\mathsf{L}(H))$$ called the Johnson homomorphism. Morita showed that the image of $\tau$ is actually contained in the smaller Lie algebra $\mathsf{Der}_\omega(\mathsf{L}(H))$ consisting of those derivations which kill the symplectic element $\omega=\sum_{i=1}^g[x_i,y_i]$. However, $\tau$ is still not onto this smaller Lie algebra, and indeed the so called Johnson cokernel $$\mathsf{C}(H)=\mathsf{Der}_\omega(\mathsf{L}(H))/\operatorname{im}(\tau)$$ is an interesting object of study. In fact $\mathsf{Der}_\omega(\mathsf{L}(H))$ is fairly well understood, so understanding $\mathsf{C}(H)$ is essentially equivalent to understanding $\mathfrak{j}(g)$!

There's a lot to say about $\mathsf{C}(H)$, but for now I'd like to highlight a theorem relating it to the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}_1$. 

Theorem?: (Matsumoto, Nakamura) There is an embedding $$\mathfrak{grt}_1\to \lim_{\dim(H)\to\infty}\mathsf{C}(H)^{\operatorname{Sp}(H)}.$$
In other words, $\mathfrak{grt}_1$ stably embeds in the $\operatorname{Sp}(H)$ invariant part of the Johnson cokernel. For each degree, it is an embedding for sufficiently high $g$. I am actually not that sure about this theorem precisely as stated. Morita's survey paper  states that a conjecture of Deligne would give an embedding of the free Lie algebra on $\sigma_{2i+1}, i\geq 1$ into the $\operatorname{Sp}$-invariant part of $\mathsf{C}(H)$, and more recently told me in an email that the recent work of F. Brown implies that there is indeed an embedding of this Lie algebra. So I assume that the theorem proven by Matsumoto and Nakamura is as stated, but again, I haven't verified this. This embedding is constructed from the absolute Galois group $\operatorname{Gal}(\overline{ \mathbb Q}/\mathbb Q)$, and perhaps in a future post I will delve into this fascinating story. The image of the embedding is often called the Galois obstruction.

Corollary: $\mathsf{L}(\sigma_3,\sigma_5,\ldots)$ embeds in $\displaystyle\lim_{\dim(H)\to\infty}\mathsf{C}(H)^{\operatorname{Sp}(H)}.$

This follows from F. Brown's theorem mentioned in a previous post. It is worthwhile to note that this is just a small part of the cokernel.

In a follow-up post I want to complete a circle of ideas by talking about recent work of my own  which defines a graph-homological invariant of  the cokernel.  Given Willwacher's theorem connecting $\mathfrak{grt}_1$ with commutative even graph homology, perhaps there is some connection between his work and mine. My invariant takes values in something constructed from the Lie graph complex, but being that the commutative and Lie operads are dual to each other, I wouldn't be surprised if there was a connection.

Next week, I will be traveling to Bonn to participate in a workshop on topological recursion and geometric quantization. I hope to blog about what I learn there, so stay tuned!

The Grothendieck-Teichmüller Lie Algebra

I'd like to talk about some recent work of Willwacher and coauthors related to the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}_1$ and graph homology. The goal of this first post is modest, just to define $\mathfrak{grt}_1$ and discuss some of what's known about it. I will be relying heavily on Willwacher's paper for my exposition.

First, we define the Drinfeld-Kohno Lie algebra $\mathfrak t_n$. It is freely generated as a Lie algebra by elements $t_{ij}$ which are symmetric in the indices $t_{ij}=t_{ji}$ where $i\neq j$ and $1\leq i,j\leq n$, subject to relations that $[t_{ij},t_{kl}]=0$ if the index sets are disjoint and
$$[t_{ij},t_{ik}]=-[t_{ij},t_{jk}], \text{ if } i,j,k \text{ are distinct indices}.$$ I encountered a similar Lie algebra in a paper of mine, where I noticed that you could interpret iterated brackets as labeled trees. Surely one can do the same here,  but with a different sign convention. For example $t_{ij}$ would be a line segment with ends labeled by $i$ and $j$. The bracket $[t_{ij},t_{ik}]$ would be a tree with one trivalent vertex and three univalent vertices labeled by $i,j,k$. I am not sure what the sign conventions should be.

Now consider the free Lie algebra on two generators, $X$ and $Y$, $\mathsf{L}(X,Y)$, and complete it to $\hat{\mathsf{L}}$. Let $\hat{\mathsf{L}}^+$ be spanned by elements of degree $3$ or higher. Consider $\varphi\in\hat{\mathsf{L}}^+$ satisfying
$$\varphi(t_{12},t_{23}+t_{24})+\varphi(t_{13}+t_{23},t_{34})=\varphi(t_{23},t_{24})+
\varphi(t_{12}+t_{13},t_{24}+t_{34})+\varphi(t_{12},t_{23}),$$ where we plug in elements of $\mathfrak{t}_4$.
The set of such $\varphi$ is the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}_1$.

There are actually two auxiliary conditions, such as the fact that $\varphi$ is antisymmetric in $X$ and $Y$, which are normally given, but they follow from a result of Furusho.

The simplest example of an element of $\mathfrak{grt}_1$ is the element $[X,[X,Y]]-[Y,[Y,X]]$.

So now that we know what $\mathfrak{grt}_1$ is, the next question is why do we care? It actually comes from a group $\mathsf{GRT}_1$, which is related to the space of Drinfeld associators with no quadratic term. It also arises in the algebraic geometry of the mapping class group and Teichmüller space, although I don't understand the story there very well. I believe that it is known that $\mathfrak{grt}_1$ embeds into the cokernel of the Johnson homomorphism as the set of so called Galois obstructions and this follows from work of Matsumoto and Nakamura.

An extremely important conjecture about $\mathfrak{grt}_1$ is the following.

Conjecture (Deligne-Mumford-Ihara): $\mathfrak{grt}_1$ is a free graded Lie algebra with generators $\sigma_3,\sigma_5,\ldots$.

F. Brown has proven half of this conjecture:

Theorem (F. Brown): The free graded Lie algebra with generators $\sigma_3,\sigma_5,\ldots$ embeds in $\mathfrak{grt}_1$.

Another remarkable result is the main theorem of Willwacher's paper. Namely, if you take Kontsevich's (commutative) even graph complex $\mathsf{GC}_2$, then

Theorem (Willwacher): There is an isomorphism $H^0(\mathsf{GC}_2)\cong \mathfrak{grt}_1$.

I am adopting Willwacher's notation here. Cohomological degree $0$ here picks up connected graphs with $E=2(V-1)$. The elements $\sigma_{2i+1}$ have a beautiful (conjectural) description as wheels with $2i+1$ spokes, which are indeed cocycles in the graph complex.

So now we have a definition of the Grothendieck-Teichmüller Lie algebra, and some idea of what is known about it and its importance. In my next posts, I intend to explore some of the ideas touched on here. For example I want to talk about $\mathsf{GC}_2$ in a lot more detail, and explore the connections to the Johnson homomorphism. I'd also like to delve into the details of Willwacher's paper.