Higher hairy graph homology

Abstract

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads $\mathsf{Comm}$ $\mathsf{Assoc}$ and $\mathsf{Lie}$ we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph homology of the trace map defined in Conant et al. (J Topology 6(1):119–153, 2013), as a symplectic representation. For the operad $\mathsf{Lie}$ assembling hairy graph cohomology classes yields all known non-trivial rational homology of $Out(F_n)$. The hairy graph homology of $\mathsf{Lie}$ is also useful for constructing elements of the cokernel of the Johnson homomorphism of a once-punctured surface.