--- title: Johnson Cokernel & Symplectic Derivations url: https://www.emergentmind.com/topics/johnson-cokernel type: topic --- # Johnson Cokernel & Symplectic Derivations The Johnson cokernel is the degreewise quotient $$ C_s = D_s(H)/\tau(\mathfrak J_s), $$ attached to the Johnson homomorphism for the mapping class group of a compact, connected, oriented surface $\Sigma_{g,1}$ of genus $g$ with one boundary component, where $H=H_1(\Sigma_{g,1};\mathbb Q)$ and $$ D_s(H)=\ker\!\big(H\otimes L_{s+1}(H)\to L_{s+2}(H)\big) $$ is the degree-$s$ piece of the Lie algebra of symplectic derivations of the free Lie algebra on $H$ [1306.3698]. It measures the gap between the Torelli graded Lie algebra and the symplectic derivation Lie algebra, and its structure is central to understanding the mapping class group and Torelli group beyond their linear actions, with deep connections to low-dimensional topology and number theory [1306.3698]. The modern theory is organized by tree models, hairy graph homology, graphical trace maps, dihedral coinvariants, and cohomology of $\mathrm{Out}(F_n)$; later work has added refined $2$-loop invariants and integral $2$-torsion phenomena in every even degree in the stable range [1610.05220][1509.03236][2508.19041][2304.12662]. ## 1. Algebraic framework Let $\Sigma_{g,1}$ be a compact, connected, oriented surface of genus $g$ with one boundary component, and write $\mathrm{Mod}_{g,1}$ for its mapping class group. The Torelli group $I_{g,1}\subset \mathrm{Mod}_{g,1}$ is the kernel of the action on $H=H_1(\Sigma_{g,1};\mathbb Q)$, where the intersection pairing gives $H$ a symplectic form with symplectic basis $\{p_1,\dots,p_g,q_1,\dots,q_g\}$ and $\omega=\sum_{i=1}^g [p_i,q_i]$ [1306.3698]. If $F=\pi_1(\Sigma_{g,1})$ and $F_{k+1}=[F,F_k]$ is the lower central series, the Johnson filtration is $$ J_s=\ker\big(\mathrm{Mod}_{g,1}\to \mathrm{Aut}(F/F_{s+1})\big), $$ with associated graded $$ \mathfrak J_s=(J_s/J_{s+1})\otimes \mathbb Q,\qquad \mathfrak J=\bigoplus_{s\ge 1}\mathfrak J_s. $$ The commutator on $\mathrm{Mod}_{g,1}$ induces a graded Lie algebra structure on $\mathfrak J$ [1306.3698]. Let $L(H)$ be the free Lie algebra on $H$, with homogeneous degree-$k$ piece $L_k(H)$. The space $$ D_s(H)=\ker\!\big(H\otimes L_{s+1}(H)\to L_{s+2}(H)\big) $$ identifies canonically with the degree-$s$ symplectic derivations $\mathrm{Der}_\omega(L(H))$, giving a Lie algebra $$ D(H)=\bigoplus_{s\ge 0} D_s(H),\qquad D^+(H)=\bigoplus_{s\ge 1}D_s(H). $$ For $\phi\in J_s$, the generalized Johnson homomorphism $$ \tau_s(\phi):H\to L_{s+1}(H),\qquad \tau_s(z)=[z^{-1}\phi(z)], $$ yields $\tau_s(\phi)\in D_s(H)$, and the total map $$ \tau:\mathfrak J\to D^+(H) $$ is a Lie algebra homomorphism. Morita proved that $\tau$ is injective, and Hain proved that $\tau(\mathfrak J)$ is the Lie algebra generated by its order-$1$ piece $\mathfrak J_1\cong D_1(H)\cong \Lambda^3H$ [1306.3698]. This framing is the standard rational setting. A later integral refinement studies the same targets over $\mathbb Z$ and shows that the cokernel contains torsion in all even degrees in the stable range, so the rational and integral theories diverge in an essential way [2304.12662]. ## 2. Tree models and hairy graph complexes A fundamental equivalent description replaces derivations by labeled trees. Let $\mathcal T_s(H)$ be the $\mathbb Q$-vector space spanned by unitrivalent trees with $s$ trivalent vertices, cyclic orderings at trivalent vertices, and leaf labels in $H$, modulo AS, IHX, and multilinearity relations. The canonical map $$ \eta_s:\mathcal T_s(H)\to H\otimes L_{s+1}(H),\qquad \eta_s(t)=\sum_x \ell(x)\otimes t_x, $$ summing over leaves $x$, induces an isomorphism $\mathcal T_s(H)\cong D_s(H)$ over characteristic zero [1306.3698]. This Lie–tree dictionary is the basis for the graphical study of the cokernel. The Conant–Kassabov–Vogtmann model enlarges trees to hairy graphs. For $C_k(H)$, one takes oriented disjoint unions of $k$ unitrivalent trees, allows some leaves to be paired by oriented external edges, and labels the remaining leaves by $H$, modulo AS, IHX, multilinearity, sign changes under reversing an external edge, and permutation signs for tree labels. The boundary $$ \partial:C_k(H)\to C_{k-1}(H) $$ contracts external edges. If $T$ denotes the operator that adds a single external edge between every pair of hairs, weighted by symplectic contraction, and $\iota:\bigwedge^k\mathcal T(H)\to C_k(H)$ is the inclusion with no external edges, the CKV trace is $$ \mathrm{Tr}^{CKV}=\exp(T)\circ \iota. $$ It is a chain map and injective on homology; in particular, the abelianization $D^{ab}(H)$ of $D^+(H)$ embeds in $H_1(C_\bullet(H))$ [1306.3698]. To pass from abelianization to the Johnson cokernel, Conant introduced a quotient that kills explicit boundaries. Let $S_2\subset C_2(H)$ be spanned by configurations where an order-$1$ tree is attached to another tree by prescribed external-edge patterns. Then $$ \Omega(H)=C_1(H)/\big(\partial(S_2)+\iota(\mathcal T(H))\big), $$ and the new trace is $$ \mathrm{Tr}^C:\mathcal T(H)\to \Omega(H), $$ defined by composing $\mathrm{Tr}^{CKV}$ with the quotient projection. The relations coming from $\partial(S_2)$ include that an isolated loop is zero, a hair may slide along an external edge, and there is a three-term relation that first appears when two or more external edges are present. The key theorem is that $\mathrm{Tr}^C$ vanishes on $\mathrm{Im}(\tau_s)$ for $s\ge 2$, so it factors through the Johnson cokernel [1306.3698]. ## 3. The Enomoto–Satoh trace and the top-level part Connected hairy graphs in $C_1(H)$ are bigraded by rank $r=b_1$ and by the number $s$ of hairs. Writing $\Omega_{s,r}(H)$ for the corresponding graded pieces, the rank-$1$ component has a concrete description: $$ \Omega_{s,1}(H)\cong [H^{\otimes s}]_{D_{2s}}, $$ where the dihedral reflection acts by $$ b(v_1\otimes\cdots\otimes v_s)=(-1)^{s+1}v_s\otimes\cdots\otimes v_1. $$ Under the identification $\mathcal T_s(H)\cong D_s(H)$, the Enomoto–Satoh trace is exactly the rank-$1$ part of the new graphical trace: $$ \frac12\,\mathrm{Tr}^{ES}=\text{projection of }\mathrm{Tr}^C\text{ onto }\Omega_{s,1}(H). $$ Thus the ES invariant is not an external add-on; it is the $1$-loop piece of the graphical trace that factors through the Johnson cokernel [1306.3698]. The rank-$1$ quotient already detects a large family. If $H^{(s)}\subset H^{\otimes s}$ is the intersection of the kernels of all pairwise contractions, then for $s>1$ and $\dim(H)\gg s$ there is an epimorphism $$ C_s\twoheadrightarrow [H^{(s)}]_{D_{2s}}, $$ with the reflection twisted by the nontrivial $\mathbb Z/2$-character when $s$ is even. This generalizes the previously known Morita and Enomoto–Satoh series. Inside these dihedral coinvariants, $[1^s]_{Sp}$ occurs if and only if $s\equiv 1\pmod 4$, with multiplicity $1$, while $[s]_{Sp}$ occurs if and only if $s$ is odd, also with multiplicity $1$; by contrast, $[s-1,1]_{Sp}$ and $[2,1^{s-2}]_{Sp}$ do not occur [1306.3698]. The multiplicity-one statements for the classical series were established explicitly by Enomoto and Satoh. In the stable range $g>k+2$, they constructed highest weight vectors in Morita’s kernel $h_{g,1}(k)$ using Brauer–Schur–Weyl duality and the Dynkin–Specht–Wever idempotent, and showed that the irreducibles $[k]$ for odd $k\ge 3$ and $[1^k]$ for $k\equiv 1\pmod 4$, $k\ge 5$, survive in the Johnson cokernel with multiplicity one [1012.2175]. The top-level part of the theory was later determined completely. Writing $$ C_d\cong C_d(d)\oplus C_d(d-2)\oplus C_d(d-4)\oplus\cdots, $$ the addendum to Conant’s paper proves that $$ Tt\circ \mathrm{Tr}_{ES}: C_d(d)\xrightarrow{\ \cong\ } [V(d)]_{D_{2d}}, $$ so the ES trace detects all top-level partitions. Equivalently, the $\mathrm{GL}(V)$-decomposition of the top-level piece is exactly the decomposition of the dihedral coinvariants of $V^{\otimes d}$ [1610.05220]. A common misconception is therefore only partially correct: the ES trace is complete on the top level, but not on the full cokernel. ## 4. Stable decompositions and explicit representation-theoretic families In the stable situation $2g=\dim(H)\gg s$, computer computations of Morita–Sakasai–Suzuki give the following low-order decompositions of the Johnson cokernel as $\mathrm{Sp}(H)$-modules [1306.3698]. | Degree $s$ | Stable decomposition of $C_s$ | |---|---| | $1$ | $0$ | | $2$ | $0$ | | $3$ | $[3]_{Sp}$ | | $4$ | $[21^2]_{Sp} \oplus [2]_{Sp}$ | | $5$ | $[5]_{Sp} \oplus [32]_{Sp} \oplus [2^2 1]_{Sp} \oplus [1^5]_{Sp} \oplus 2[21]_{Sp} \oplus 2[1^3]_{Sp} \oplus 2[1]_{Sp}$ | | $6$ | $2[41^2]_{Sp} \oplus [3^2]_{Sp} \oplus [321]_{Sp} \oplus [31^3]_{Sp} \oplus [2^2 1^2]_{Sp} \oplus 2[4]_{Sp} \oplus 3[31]_{Sp} \oplus 3[2^2]_{Sp} \oplus 3[21^2]_{Sp} \oplus 2[1^4]_{Sp} \oplus [2]_{Sp} \oplus 5[1^2]_{Sp} \oplus 3[0]_{Sp}$ | The rank-$1$ detection accounts for the full size-$s$ part up to $s=6$: $$ [H^{(4)}]_{D_8}\cong [21^2]_{Sp},\qquad [H^{(5)}]_{D_{10}}\cong [5]_{Sp}\oplus [32]_{Sp}\oplus [2^2 1]_{Sp}\oplus [1^5]_{Sp}, $$ and $$ [H^{(6)}]_{D_{12}}\cong [3^2]_{Sp}\oplus 2[41^2]_{Sp}\oplus [321]_{Sp}\oplus [31^3]_{Sp}\oplus [2^2 1^2]_{Sp}. $$ Comparing with the table shows that these dihedral coinvariants account for all size-$s$ irreducibles present in the computed cokernels up to $s=6$ [1306.3698]. The representation-theoretic analysis uses Schur–Weyl duality and dihedral character averages. For an irreducible symmetric-group representation $[\lambda]_{\mathfrak S_s}$ with character $\chi_\lambda$, $$ \dim\big(([\lambda]_{\mathfrak S_s})_{D_{2s}}\big)=\frac1{2s}\sum_{g\in D_{2s}}\chi_\lambda(g), $$ and with the $\mathbb Z/2$-twist, $$ \dim=\frac1{2s}\sum_{g\in D_{2s}}\sigma(g)\chi_\lambda(g), $$ where $\sigma(a)=+1$ on rotations and $\sigma(b)=-1$ on reflections [1306.3698]. This yields explicit infinite families. For $p\ge 3$ prime, if $\lambda=(k,p-k)$ and $$ \alpha_k=\binom pk-\binom p{k-1}, $$ then $[k,p-k]_{Sp}$ appears in $C_p$ with multiplicity $\alpha_k/(2p)$ when $k>1$ is odd. If $k=2m$ is even and $$ \beta_m=\binom{(p-1)/2}{m}-\binom{(p-1)/2}{m-1}, $$ then $[k,p-k]_{Sp}$ appears in $C_p$ with multiplicity $(\alpha_{2m}+\beta_m)/2$. For $s=2p$ with $p\ge 3$ prime and $1\ell\ge 0$, where $\mathcal S_w$ and $\mathcal M_w$ are cusp forms and modular forms of weight $w$ [1306.3698]. A cohomological reformulation places these invariants in the top cohomology of $\mathrm{Out}(F_n)$ with twisted coefficients. For a cocommutative Hopf algebra $H$, there is a natural $\mathrm{Aut}(F_n)$-action on $H^{\otimes n}$ inducing an $\mathrm{Out}(F_n)$-action on a quotient $\overline{H^{\otimes n}}$. In the case $H=T(V)$, the invariant $\mathrm{Tr}^C$ projects to $$ H^{2n-3}\big(\mathrm{Out}(F_n);\overline{T(V)^{\otimes n}}\big), $$ and for all $n\ge 2$, in the stable range in genus, this gives an invariant defined on the Johnson cokernel taking values in that cohomology group; moreover, for large enough $g$ compared to $|\lambda|$, it surjects onto each $\mathrm{Sp}(V)$-type $[\lambda]_{Sp}$ inside the $\mathrm{GL}(V)$-type $[\lambda]_{GL}$ contained in the image [1509.03236]. The case $n=2$ is especially explicit because $\mathrm{Out}(F_2)\cong \mathrm{GL}_2(\mathbb Z)$. There one obtains $$ H^1\big(\mathrm{GL}_2(R);\overline{\mathrm{Sym}(\mathfrak L_{(2)}\otimes R^2)}\big) \cong \bigoplus_{k>\ell\ge 0}\Big(\mathcal S_{2k-2\ell+2}\otimes \overline{(2k,2\ell)}(\mathfrak L_{(2)})\Big) \oplus \bigoplus_{k>\ell\ge 0}\Big(\mathcal M_{2k-2\ell+2}\otimes \overline{(2k+1,2\ell+1)}(\mathfrak L_{(2)})\Big), $$ which yields explicit infinite families of obstructions in the Johnson cokernel [1509.03236]. ## 6. Integral torsion and refined $2$-loop detection The rational theory does not exhaust the subject. In the integral setting, Faes studies the Satoh trace $$ \mathrm{Tr}: \mathrm{Der}(H)\to C(H)=T(H)/[T(H),T(H)] $$ and defines, for each $k\ge 2$, an $\mathrm{Sp}(H)$-equivariant obstruction $$ \overline{\mathrm{Tr}}_k:\ker(\mathrm{Tr}_k)\cap D_k(H)\to C_{k+1}(H)/\overline{C_{k+1}(H)}, $$ which is independent of the symplectic expansion, vanishes on $\mathrm{Im}(\tau_k)$, and has $2$-torsion image. After quotienting further by the mirror subgroup $\mathrm{Mir}_{k+1}(H)$ one gets $$ \mathrm{Tr}^{Mir}_k:\ker(\mathrm{Tr}_k)\cap D_k(H)\to C_{k+1}(H)/\mathrm{Mir}_{k+1}(H), $$ which is nontrivial for even $k$ and vanishes for odd $k$ [2304.12662]. The main consequence is integral torsion in every even degree in the stable range. For any $k\ge 1$ and $n\ge 2k+4$, equivalently $g\ge k+2$, the cokernel $$ D_{2k}(H)/\mathrm{Im}(\tau_{2k}) $$ has nontrivial $2$-torsion [2304.12662]. This shows that even after accounting for the known rational traces, the image of the Johnson homomorphisms is integrally smaller than $\ker(\mathrm{Tr})\cap D_k(H)$. A different refinement concerns the failure of the $1$-loop trace to capture the full rational cokernel in degree $6$. In the stable range, the Johnson cokernel decomposes by loop number $$ \mathrm{cok}(n)\cong \bigoplus_{r=1}^{\lfloor n/2\rfloor+1}\widetilde{\Omega}_{r,\langle n+2-2r\rangle}, $$ where the $1$-loop part is the ES obstruction. The ES trace is injective for $n\le 5$, but at $n=6$ one has $$ \big(h(6)\cap \ker \widetilde{\mathrm{Tr}}_1\big)/m(6)\cong [1^4]_{Sp}\oplus [1^2]_{Sp}\oplus [0]_{Sp}. $$ The refined $2$-loop trace remedies this: for $g$ sufficiently large, $$ \widetilde{\mathrm{Tr}}_1\oplus \widetilde{\mathrm{Tr}}_2:\mathrm{cok}(6)\to \widetilde{\Omega}_{1,6}\oplus \widetilde{\Omega}_{2,4} $$ is injective, and $\widetilde{\mathrm{Tr}}_2$ captures the $[1^4]_{Sp}$, $[1^2]_{Sp}$, and $[0]_{Sp}$ components invisible to $\widetilde{\mathrm{Tr}}_1=\mathrm{Tr}^{ES}$ [2508.19041]. This also clarifies a subtle point about earlier trace maps. Conant’s original $2$-loop trace $Tr_2^C$ factors through the $1$-loop trace: $$ \Phi\circ Tr_1^C = 3\cdot Tr_2^C, $$ so it yields no new obstructions beyond the ES trace. By contrast, the refined $\widetilde{\mathrm{Tr}}_2$ does not factor through $\widetilde{\mathrm{Tr}}_1$ and detects genuinely new degree-$6$ components [2508.19041]. ## 7. Rigidity, misconceptions, and open directions The graphical-trace viewpoint has both strength and rigidity. In the ribbon-graph framework of Merkulov–Willwacher, the ES trace arises from the unique ribbon graph with one vertex and one edge, and this graph maps to the graded Turaev cobracket $\delta^{gr}$. Taniguchi proves that the space of $1$-cocycles in the ribbon graph complex with respect to the vertex grading is one-dimensional and spanned by this graph. Consequently, within that ribbon-graph complex there are no other linearly independent $1$-cocycles that could yield new trace-type invariants annihilating the Johnson image [2507.20492]. In that precise sense, the ES trace is the only source of detection there. Several limitations remain explicit in the current literature. Rank-$1$ detection via dihedral coinvariants appears to account for all size-$s$ components up to $s=6$, and Conant states this conjecturally for all $s$; however, higher ranks $r\ge 2$ remain only partially understood [1306.3698]. The refined $2$-loop theory provides a complete degree-$6$ detection beyond ES, but higher-loop parts and a full stable $\mathrm{Sp}$-decomposition of all cokernels remain open [2508.19041]. On the integral side, Faes asks whether $\overline{C_k(H)}=\mathrm{Mir}_k(H)$ for $n\gg k$ integrally; rationally the quotient is torsion, but the integral equality remains open [2304.12662]. The resulting picture is therefore stratified rather than uniform. The ES trace completely controls the top-level partitions [1610.05220]; it does not, however, determine the full cokernel, as degree $6$ already requires refined $2$-loop data [2508.19041]. Rationally, hairy-graph and $\mathrm{Out}(F_n)$-cohomological methods detect large families and modular-form phenomena [1509.03236]; integrally, new $2$-torsion obstructions persist in every even degree in the stable range [2304.12662]. The Johnson cokernel is thus best viewed as a multi-layered object whose rank-$1$, higher-loop, cohomological, and torsion aspects are complementary rather than interchangeable. Source: https://www.emergentmind.com/topics/johnson-cokernel