Graphical obstructions to the surjectivity of the higher order Johnson homomorphism

The Johnson filtration of the mapping class group gives rise to an associated graded Lie algebra. The higher order Johnson homomorphism embeds this Lie algebra in the larger but better understood Lie algebra of symplectic derivations. It has long been known that this is not surjective, and various families of obstructions to surjectivity have been constructed by several authors. We give a graph homology construction that unifies and expands all of the previously known obstructions (with one exception, the so-called Galois obstruction.)