The Johnson cokernel and the Enomoto–Satoh invariant

Abstract

ABSTRACT. We study the cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto-Satoh and Conant-Kassabov-Vogtmann is given, and using technology from the author's work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes series due to Morita and Enomoto-Satoh. The Enomoto-Satoh trace is the rank 1 part of the new trace. The rank 2 part is also investigated.

Hopf algebras and invariants of the Johnson cokernel

“…In [5] a "trace" map Tr C : D + → ⊕ n≥1 Ω n (V) was constructed. The spaces Ω n (V) are generated by graphs formed by adding n "external" oriented edges to a labeled tree from D + , while the trace map is defined by adding sets of edges to a tree in all possible ways, multiplying by contractions of labeling coefficients.…”

“…In fact one recovers the abelianization obstructions from Theorem 1.1 by applying the map T(V) → Sym(V) to the coefficient modules. In obstruction modules from [5]:…”

“…All of these obstructions are organized in Figure 1. The bottom row is given by the abelianization obstructions and the top by the obstructions from [5]. Sitting in between these are the new obstruction modules which are the subject of this paper.…”

An Abelian Quotient of the Symplectic Derivation Lie Algebra of the Free Lie Algebra

“…a 3 , a 4 , a 5 , b 2 , b 5 , b 3 , a 6 , a 7 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b3 , b 5 , b 6 , b 4 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 3 , b 5 , a 7 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 3 , b 6 , b 5 , b 4 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 3 , a 7 , b 5 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 3 , a 7 , b 5 , b 6 , b 4 , b 7 },{a 2 , a 3 , a 4 , a 5 , b 2 , b 4 , a 6 , b 3 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , b 4 , a 6 , b 3 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 4 , b 3 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 4 , b 3 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , b 6 , b 3 , b 4 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 2 , a 6 , a 7 , b 3 , b 4 , b 5 , b 6 , b 7 },…”

“…a 3 , a 4 , a 5 , b 3 , b 2 , b 4 , a 6 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , b 4 , a 6 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b3 , b 2 , a 6 , b 4 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , a 6 , b 4 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , a 6 , a 7 , b 4 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , a 6 , a 7 , b 4 , b 6 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , b 5 , a 6 , b 6 , b 4 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , b 5 , a 6 , a 7 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , a 6 , b 5 , b 6 , b 4 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 3 , b 2 , a 6 , b 5 , a 7 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 4 , b 2 , b 3 , a 6 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 4 , b 2 , b 3 , a 6 , a 7 , b 5 , b 6 , b 7 },{a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 4 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 4 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 5 , b 4 , b 6 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 6 , b 4 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , a 7 , b 4 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , a 7 , b 4 , b 6 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 5 , b 2 , b 3 , a 6 , a 7 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 5 , b 6 , b 4 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 5 , a 7 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , b 6 , b 5 , b 4 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , a 7 , b 5 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 3 , a 7 , b 5 , b 6 , b 4 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 4 , b 2 , a 6 , b 3 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 4 , b 2 , a 6 , b 3 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 4 , b 3 , b 6 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 4 , b 3 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , b 6 , b 3 , b 4 , b 5 , a 7 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 2 , a 7 , b 3 , b 4 , b 5 , b 6 , b 7 },…”