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.