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The Johnson homomorphism, embedding calculus and graph complexes

Florian Naef, Thomas Willwacher

TL;DR

The paper connects the Johnson homomorphism and its cokernel to graph complexes arising from Goodwillie-Weiss embedding calculus, providing a cohesive framework that unifies these invariants with higher-loop Enomoto-Satoh traces. It proves stable injectivity of the Johnson homomorphism in weight regimes $g\ge 3W\ge 3$, and gives an explicit, representation-theoretic decomposition of the cokernel in terms of $\mathrm{Sp}(2g)$-representations with multiplicities $M_{\mu}^W$, computed via top-weight Euler characteristics $\chi_{h,\lambda}$ of moduli spaces and branching data. The approach leverages a loop-order filtration and a spectral sequence to identify higher ES traces whose images detect the cokernel, and it provides concrete calculations up to weight $6$, showing where higher-loop contributions first appear. Finally, the work situates these graph complexes within the embedding calculus framework, offering a conceptual link to diffeomorphism groups and the rational Torelli-like structure of surface automorphisms.

Abstract

We explain how the Johnson homomorphism and the Enomoto-Satoh trace, as well as higher-loop-order generalizations, can be obtained from graph complexes originating in the Goodwillie-Weiss calculus. This paper can be seen as an addendum to our earlier work. It contains little new mathematical content, but is intended to give an overview of a different viewpoint on the Johnson homomorphism, for experts working mainly in the latter area.

The Johnson homomorphism, embedding calculus and graph complexes

TL;DR

The paper connects the Johnson homomorphism and its cokernel to graph complexes arising from Goodwillie-Weiss embedding calculus, providing a cohesive framework that unifies these invariants with higher-loop Enomoto-Satoh traces. It proves stable injectivity of the Johnson homomorphism in weight regimes , and gives an explicit, representation-theoretic decomposition of the cokernel in terms of -representations with multiplicities , computed via top-weight Euler characteristics of moduli spaces and branching data. The approach leverages a loop-order filtration and a spectral sequence to identify higher ES traces whose images detect the cokernel, and it provides concrete calculations up to weight , showing where higher-loop contributions first appear. Finally, the work situates these graph complexes within the embedding calculus framework, offering a conceptual link to diffeomorphism groups and the rational Torelli-like structure of surface automorphisms.

Abstract

We explain how the Johnson homomorphism and the Enomoto-Satoh trace, as well as higher-loop-order generalizations, can be obtained from graph complexes originating in the Goodwillie-Weiss calculus. This paper can be seen as an addendum to our earlier work. It contains little new mathematical content, but is intended to give an overview of a different viewpoint on the Johnson homomorphism, for experts working mainly in the latter area.
Paper Structure (10 sections, 7 theorems, 45 equations)

This paper contains 10 sections, 7 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Three dg Lie algebras of graphs -- recollections from FNW
  3. Definition
  4. Stable cohomology
  5. The loop filtration on graph complexes and the Johnson homomorphism
  6. Definition and stable injectivity
  7. The spectral sequence and higher Enomoto-Satoh traces
  8. The Johnson cokernel and proof of Theorem \ref{['thm:intro euler']}
  9. Explicit example: The Johnson cokernel and ES traces in weights $\leq 6$
  10. Concluding remarks: Diffeomorphism groups and the embedding calculus

Key Result

Theorem 1

[Corollary cor:injective below] The Johnson homomorphism $J_{g,1}$ is injective in weights $W$ such that $g\geq 3W\geq 3$.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Lemma 3: FNW
  • Theorem 4: FNW, see also KRWnew
  • Definition 5
  • Proposition 6
  • proof
  • Corollary 7
  • Definition 8
  • Corollary 9
  • ...and 1 more