The Johnson Cokernel and the Enomoto-Satoh invariant

Addendum to "The Johnson cokernel and the Enomoto-Satoh invariant": The ES-trace detects all top-level partitions

The degree $d$ part of the cokernel $\mathsf C_d$ of the Johnson homomorphism decomposes into irreducible $\mathrm{SP}$-modules indexed by partitions of $d-2r$ for $r\geq 0$: $$\mathsf C_d\cong \mathsf C_d(d)\oplus \mathsf C_d(d-2)\oplus\cdots.$$ In this addendum we calculate $\mathsf{C}_d(d)$ precisely: it is isomorphic to the $\mathrm{GL}(V)$-decomposition of a space of coinvariants $(V^{\otimes d})_{D_{2d}}$, and the isomorphism is induced by Enomoto and Satoh's trace map. This establishes Conjecture 7.2 of the paper "The Johnson Cokernel and the Enomoto-Satoh invariant."

关于约翰逊余核的2环部分

On the 2-loop part of the Johnson cokernel

本文研究约翰逊余核的2环部分，通过2环迹映射揭示6度中无法被Enomoto-Satoh迹检测到的余核组成部分。

约翰逊同态、嵌入微分和图复形

The Johnson homomorphism, embedding calculus and graph complexes

本文通过图复形解释约翰逊同态及更高阶扩展的起源，为专家提供不同视角的概述。

Hopf Algebras and Invariants of the Johnson Cokernel

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case,…