The Lie Lie algebra

We study Kontsevich's Lie algebra h+ associated with the Lie operad. We study two functors on cocommutative Hopf algebras H_i(H) and Omega_i(H). The first H_i(H)= H^{vcd}(Out(F_i);\bar{H^{\otimes i}}) generalizes hairy graph homology. When H is the symmetric algebra Sym(V), it determines the abelianization of h+. When H is a quotient of the tensor algebra T(V) it gives invariants of the cokernel of the Johnson homomorphism. The functor Omega_i(H) generalizes an invariant for the Johnson cokernel given in earlier work, and gives Johnson cokernel invariants when H is a quotient of T(V). By work of Morita and Conant-Kassabov-Vogtmann, H_i(Sym(V)) is known for i<=2. In this paper, we compute a presentation for H_3(Sym(V)), which allows us to show the even degree part is quite large, and to make low degree computations in the odd degree case. Moving on to the modules Omega_i(H), the module Omega_1(T(V))=\oplus_k(V^{\otimes k})_{D_{2k}} is the target of the Enomoto-Satoh trace map. In this paper we make complete calculations for Omega_2(H) for H=Sym(V) and H=U(\mathsf L_{(2)}(V)), where U(\mathsf L_{(2)}) is the universal enveloping algebra of the free nilpotent Lie algebra of nilpotency class 2, giving higher level Johnson cokernel obstructions than the Enomoto-Satoh invariant. In the last section we summarize low degree computer computations for H_i(H) and Omega_i(H) for i=2, 3.