The first Galois obstruction in the Johnson cokernel

Abstract: We explicitly determine the first Galois obstruction in the cokernel of the Johnson homomorphism of the mapping class group of a surface of genus $g$, for every genus $g \ge 2$. It is described as a sum of two terms which are considerably different in character. One lies in the kernel of the Enomoto-Satoh trace map, whereas the other belongs to a certain ideal which vanishes upon passage to the closed surface case.

• The paper explicitly determines the first Galois obstruction for every genus g≥2 as σ₃(g)=180(g−1)gε−j, separating a trace-kernel component from a boundary-dependent correction term.
• Using symplectic representation theory, chord-diagram calculations, detector functionals, and polynomial interpolation in the genus, the authors prove that this representative normalizes the Johnson image.
• The paper computes the trace ratio ES₆¹(σ₃(g))/ES₆²(σ₃(g))=2g−1, showing that the obstruction’s nontrivial Enomoto–Satoh trace comes entirely from the correction term j.

The paper determines explicitly the first Galois obstruction $\sigma_3(g)=\hat{\tau}_{g,1}(6)(\sigma_3)$ inside the cokernel of the Johnson homomorphism for surfaces of every genus $g\ge 2$, and computes the value of the Enomoto–Satoh trace map on it. The obstruction is expressed as a sum of two structurally distinct terms: one lying in the kernel of the Enomoto–Satoh trace map and the other in the Lie ideal that vanishes upon passage to the closed-surface setting.

Background: Johnson homomorphisms, Galois obstructions, and the Enomoto-Satoh trace

Let $\Sigma_{g,1}$ be a genus-$g$ oriented surface with one boundary component. The action of the mapping class group $\mathcal{M}_{g,1}$ on the free group $\pi_1(\Sigma_{g,1})$ of rank $2g$ yields the Johnson filtration $\{\mathcal{M}_{g,1}(k)\}$, and the $k$-th Johnson homomorphism $\tau_{g,1}^Z(k)$ embeds the associated graded Lie algebra into $g\ge 2$0, the graded Lie algebra of symplectic derivations of the free Lie algebra generated by $g\ge 2$1. Two classical problems are to determine the image of $g\ge 2$2 and to give a topological meaning to its cokernel.

Two objects address these problems from different directions. First, the Enomoto–Satoh trace map $g\ge 2$3 vanishes identically on the Johnson image, hence detects cokernel elements (2608.17673). Second, Hain's graded arithmetic Johnson homomorphism injects the motivic Lie algebra $g\ge 2$4 — free on generators $g\ge 2$5 of degree $g\ge 2$6 — into the $g\ge 2$7-invariant part of the Johnson cokernel; its image consists of the Galois obstructions, whose existence was conjectured by Oda and proved by Nakamura, Matsumoto, and Takao, with injectivity resting on Brown's theorem. These elements are well-defined only up to nonzero scalar and modulo the Johnson image. Degree 6 is the first degree where the two theories meet: $g\ge 2$8 is defined exactly in degrees matching the Galois obstructions.

A key structural fact, established via the Goldman–Turaev framework of Kawazumi–Kuno and Alekseev–Kawazumi–Kuno–Naef as surveyed in Hain's work, is that the Turaev cobracket kernel coincides with the kernel of the Enomoto–Satoh trace map; since the Turaev cobracket of $g\ge 2$9 is nontrivial for all $\Sigma_{g,1}$0 and $\Sigma_{g,1}$1, one has $\Sigma_{g,1}$2. The present paper makes this nontriviality explicit in the first case $\Sigma_{g,1}$3.

Main results

Prior results of the authors give a short exact sequence

$\Sigma_{g,1}$4

with $\Sigma_{g,1}$5 ($\Sigma_{g,1}$6) or $\Sigma_{g,1}$7 ($\Sigma_{g,1}$8), and a direct sum decomposition

$\Sigma_{g,1}$9

where $g$0 is the ideal vanishing under projection to the pointed closed surface. The arithmetic Johnson map descends to $g$1, so the first Galois obstruction lies in $g$2, and its $g$3-component must be a nonzero scalar multiple of an "extra element" $g$4 completing the Johnson basis.

The main theorem gives, for the normalized basis $g$5 (with $g$6 at $g$7):

$g$8

where

$g$9

and $\mathcal{M}_{g,1}$0. The two summands differ fundamentally in character: $\mathcal{M}_{g,1}$1 is written in normalized (i-stable) coordinates while $\mathcal{M}_{g,1}$2 is written in ordinary (p-stable) coordinates.

The second theorem computes the trace value:

$\mathcal{M}_{g,1}$3

where $\mathcal{M}_{g,1}$4 are the multi-contraction components corresponding to the permutations $\mathcal{M}_{g,1}$5 and $\mathcal{M}_{g,1}$6. Since $\mathcal{M}_{g,1}$7, this ratio is entirely determined by the correction term $\mathcal{M}_{g,1}$8. This ratio is well-defined because both the obstruction's projective ambiguity and the vanishing of $\mathcal{M}_{g,1}$9 on the Johnson image leave the projective class of $\pi_1(\Sigma_{g,1})$0 invariant. A consequence is that the entire nontriviality of $\pi_1(\Sigma_{g,1})$1 on the first Galois obstruction arises through the ideal $\pi_1(\Sigma_{g,1})$2: passing to the quotient by this ideal, $\pi_1(\Sigma_{g,1})$3 and the normalizer $\pi_1(\Sigma_{g,1})$4 coincide.

Method

The proof exploits Hain's observation that the Galois obstructions normalize the extended Johnson image. In the semidirect product $\pi_1(\Sigma_{g,1})$5, the normalizer $\pi_1(\Sigma_{g,1})$6 is a trivial $\pi_1(\Sigma_{g,1})$7-module, and $\pi_1(\Sigma_{g,1})$8 is generated by $\pi_1(\Sigma_{g,1})$9.

The authors construct a basis of $2g$0 using tabulated values of $2g$1 on the chord-diagram basis for genera $2g$2; dividing by Casimir eigenvalues produces genus-independent values on the normalized basis, which identifies the extra element $2g$3.

To measure the failure of $2g$4 to normalize the Johnson image, they analyze the $2g$5- and $2g$6-isotypical parts of $2g$7. Using explicit detectors $2g$8, they establish exact sequences showing that the $2g$9-component of the degree-7 Johnson image is $\{\mathcal{M}_{g,1}(k)\}$0, the $\{\mathcal{M}_{g,1}(k)\}$1-component is $\{\mathcal{M}_{g,1}(k)\}$2, and $\{\mathcal{M}_{g,1}(k)\}$3 contains one additional copy of each, contained in $\{\mathcal{M}_{g,1}(k)\}$4. This yields seven detecting functionals $\{\mathcal{M}_{g,1}(k)\}$5 on $\{\mathcal{M}_{g,1}(k)\}$6: membership of $\{\mathcal{M}_{g,1}(k)\}$7 in $\{\mathcal{M}_{g,1}(k)\}$8 is equivalent to $\{\mathcal{M}_{g,1}(k)\}$9, and normalization of the Johnson image further requires $k$0.

A central technical step proves that all these functionals are polynomials in $k$1 of degree at most 3, so four computed genera determine them universally. This rests on an explicit combinatorial identity showing that brackets of highest-weight vectors with arbitrary linear-chord-diagram tensors reduce to permutations of fixed tensors, combined with the fact that triple contractions of $k$2 are monomials of degree $k$3 in $k$4. Direct evaluation then gives

$k$5

so that $k$6 kills both functionals and therefore normalizes the Johnson image; uniqueness up to scalar and addition of Johnson-image elements identifies it with $k$7.

It should be noted that the identification of $k$8 across genera via chord diagrams is used throughout, and that the polynomiality argument relies on computer-assisted computation (Mathematica) for five genera together with the Vandermonde interpolation principle; the paper also concedes that the case $k$9 requires separate treatment since $\tau_{g,1}^Z(k)$0 vanishes and some functionals are undefined there, though the argument adapts with minor modification.

Open questions raised

The paper poses several specific problems. Whether the Turaev cobracket of the correction term $\tau_{g,1}^Z(k)$1 can be computed directly is left open. More significantly, the authors define a reduced Enomoto–Satoh trace map $\tau_{g,1}^Z(k)$2 on the closed-surface target and conjecture that $\tau_{g,1}^Z(k)$3 for all $\tau_{g,1}^Z(k)$4 and $\tau_{g,1}^Z(k)$5 — verified here for $\tau_{g,1}^Z(k)$6. If true, no known obstruction currently distinguishes the arithmetic Johnson image from the geometric Johnson image over closed surfaces, and the meaning of the $\tau_{g,1}^Z(k)$7-valued "correction term" would require interpretation. Further open questions concern the relation among $\tau_{g,1}^Z(k)$8 across genera for fixed $\tau_{g,1}^Z(k)$9, and effective methods for computing the abelianization $g\ge 2$00, noting that any abelianization class outside $g\ge 2$01 must come from the Johnson cokernel.

Conclusion

This paper provides the first explicit representative of a Galois obstruction inside the Johnson cokernel of the boundary mapping class group, valid for all $g\ge 2$02. The decomposition $g\ge 2$03 separates an i-stable term detected by the Enomoto–Satoh trace machinery from a p-stable correction living in the boundary-killing ideal, and the exact ratio $g\ge 2$04 pins down the trace value completely. The result suggests, but does not prove, that after reducing modulo $g\ge 2$05 all Galois obstructions lie in the kernel of the Enomoto–Satoh trace map — a question whose resolution would clarify whether new obstructions are needed to detect arithmetic phenomena in the closed-surface Johnson cokernel.