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!