Higher hairy graph homology

TL;DR

This paper explores the structure of hairy graph homology related to cyclic operads, connecting it to graph cohomology, dihedral homology, and applications to Out(F_n) and surface mapping class groups.

Contribution

It introduces methods to assemble hairy graph cohomology classes into graph cocycles, computes associated dihedral homology for key operads, and links hairy graph homology to known homological invariants.

Findings

Hairy graph homology classes can be assembled into graph cocycles.

Explicit computation of dihedral homology for Comm, Assoc, and Lie operads.

Identification of the image of the trace map as a symplectic representation.

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 Comm, Assoc and 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 [CKV], as a symplectic representation. For the operad Lie assembling hairy graph cohomology classes yields all known non-trivial rational homology of Out(F_n). The hairy graph homology of Lie is also useful for constructing elements of the cokernel of the Johnson homomomorphism of a once-punctured surface.