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

Hopf algebras and invariants of the Johnson cokernel

“…In the case when H = T(V) is the tensor algebra, we show that the invariant Tr C of the cokernel of the Johnson homomorphism studied in [5] projects to take values in H vcd (Out(F n ); H ⊗n ). We analyze the n = 2 case, getting large families of obstructions generalizing the abelianization obstructions of [8].arXiv:1509.03236v1 [math.AT] 10 Sep 2015where GL 2 (Z) acts on Sym(L (2) ) ⊗2 ∼ = Sym(L (2) ⊗ k 2 ) via the standard action on k 2 , which induces the action of GL 2 (Z) on Sym(L (2) ) ⊗2 . This latter group can be computed via the methods of [7], leading to many families of obstructions [λ] Sp ⊗ M k and [λ] Sp ⊗ S k , where M k , S k denote spaces of modular (respectively cusp) forms of weight k. The simplest new families of obstructions one gets are [2k − 1, 1 2 ] Sp ⊗ S 2k+2 ⊂ C 2k+5 , and ([2k + 1, 1 2 ] Sp ⊕ [2k, 2, 1] Sp ⊕ [2k, 1 3 ] Sp ) ⊗ M 2k+2 ⊂ C 2k+7 .…”

“…For all n ≥ 1, the chain complex G (n)Lie V is quasi-isomorphic to the chain complex G (n) Sym(V)Lie .Proof. See section 5.3 of[8].…”

The Cohomology of the Full Directed Graph Complex

“…Using this isomorphism, we easily obtain the following generalizations of statements of Theorems 3.1 and 3.2: Theorem 3.7 Let d be any even integer and v 4m+1−d be a symbol of degree 4m + 1 − d. The natural embeddings [2], [6], [8], [16], [17], [27], [28], [29] for more details about these families of graph complexes and their generalizations. For odd d, the directions on edges play a special role.…”

Commutative hairy graphs and representations of Out (Fr)

“…, m N in R n can be expressed through the graph cohomology of a hairy graph complex HGC m1,...,mN ;n , generalizing the complex HGC m,n arising in the case N = 1. Similar graph-complexes were also considered in [5,6]. † The complex HGC m1,...,mN ;n differs in so far that hairs are N -colored, with the j-colored hairs carrying cohomological degree † In these works the construction is more general on internal vertices, allowing any cyclic operad as input (commutative operad in our case), but slightly more restrictive on the hair vertices, allowing only even number of colors of the same degrees -which are basis elements of a symplectic vector space.…”