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On the 2-loop part of the Johnson cokernel

Yusuke Kuno, Masatoshi Sato

TL;DR

This work advances the understanding of the Johnson cokernel by isolating and explicitly presenting its 2-loop component via the hairy Lie graph complex. It provides a concrete presentation of the 2-loop space $\widetilde{\Omega}_2$ through generators $\Theta(t,u,v,w)$ and the generalized Conant third relation, and uses this to analyze the degree-6 Johnson cokernel, identifying specific $\mathrm{GL}$- and $\mathrm{Sp}$-modules that appear. The paper also clarifies how two different 2-loop trace constructions relate, showing that any 2-loop information captured beyond the Enomoto–Satoh trace is already detected by ES, up to a proved scalar relation between traces. Overall, the results connect graph-complex models of the Johnson cokernel with explicit module decompositions, deepening the link between low-dimensional topology, representation theory, and number-theoretic phenomena.

Abstract

We study stable Sp-decompositions of the cokernel of the Johnson homomorphism. Continuing the work of Conant in 2016, which identified the 1-loop part of the Johnson cokernel as the Enomoto-Satoh obstruction, we study the 2-loop part. Using the corresponding 2-loop trace map, we capture all the components of the Johnson cokernels in degree 6 that cannot be detected by the Enomoto-Satoh trace.

On the 2-loop part of the Johnson cokernel

TL;DR

This work advances the understanding of the Johnson cokernel by isolating and explicitly presenting its 2-loop component via the hairy Lie graph complex. It provides a concrete presentation of the 2-loop space through generators and the generalized Conant third relation, and uses this to analyze the degree-6 Johnson cokernel, identifying specific - and -modules that appear. The paper also clarifies how two different 2-loop trace constructions relate, showing that any 2-loop information captured beyond the Enomoto–Satoh trace is already detected by ES, up to a proved scalar relation between traces. Overall, the results connect graph-complex models of the Johnson cokernel with explicit module decompositions, deepening the link between low-dimensional topology, representation theory, and number-theoretic phenomena.

Abstract

We study stable Sp-decompositions of the cokernel of the Johnson homomorphism. Continuing the work of Conant in 2016, which identified the 1-loop part of the Johnson cokernel as the Enomoto-Satoh obstruction, we study the 2-loop part. Using the corresponding 2-loop trace map, we capture all the components of the Johnson cokernels in degree 6 that cannot be detected by the Enomoto-Satoh trace.

Paper Structure

This paper contains 23 sections, 16 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The Johnson cokernel and its loop decomposition
  3. Main results
  4. Description of the $2$-loop space
  5. The Johnson cokernel in degree $6$
  6. Relation between two $2$-loop traces
  7. Organization
  8. Notation
  9. Hairy Lie graph complex and the Johnson cokernel
  10. Hairy Lie graph complex
  11. Conant's description of the Johnson cokernel
  12. Description of the $2$-loop space
  13. Description of $C_{1,2}\mathcal{H}$
  14. Generalized Conant's third relation
  15. Proof of Theorem \ref{['thm:omega2']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

The space $\widetilde{\Omega}_2$ is generated by elements $\Theta(t,u,v,w)$ for all $t,u,v,w \in \mathcal{T}$ subject to the following relations:

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Morita, Sakasai and Suzuki
  • Remark 1.4
  • Proposition 1.5: $=$ Proposition \ref{['prop:factorthrough']}
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Conant Con16, Theorem 2.1
  • ...and 23 more