On the 2-loop part of the Johnson cokernel

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.