Hairy graphs to ribbon graphs via a fixed source graph complex

Abstract

We show that the hairy graph complex (HGC n,n , d) appears as an associated graded complex of the oriented graph complex (OGC n+1 , d), subject to the filtration on the number of targets, or equivalently sources, called the fixed source graph complex. The fixed source graph complex (OGC 1 , d 0 ) maps into the ribbon graph complex RGC, which models the moduli space of Riemann surfaces with marked points. The full differential d on the oriented graph complex OGC n+1 corresponds to the deformed differential d + h on the hairy graph complex HGC n,n , where h adds a hair. This deformed complex (HGC n,n , d + h) is already known to be quasi-isomorphic to standard Kontsevich's graph complex GC 2 n . This gives a new connection between the standard and the oriented version of Kontsevich's graph complex.

Oriented hairy graphs and moduli spaces of curves

“…In the case that the composition of the cobracket and bracket To simplify signs, we shall also consider a graded version, ΛLieB, for which the bracket and cobracket both have cohomological degree +1, and are symmetric operations. More precisely, a ΛLieB-structure on the graded vector space V consists of a Lie algebra structure on V [1], and a Lie coalgebra structure on V[−1], which satisfy a graded version of the Drinfeld five-term identity. Similarly, we may consider the graded versions of the properad of involutive Lie bialgebras ΛILieB and the corresponding resolutions ΛLieB ∞ , ΛILieB ∞ .…”

“…More precisely, if |S | > 1 the mapping cone is acyclic. For |S | = 1, say S = [1], the mapping cone has one-dimensional cohomology, spanned by the graph (10) 1 ∈ HGC [1] n in genus 1. We refer to the proof of [22,Proposition 3.4] for the detailed argument.…”

“…The biggest difference is that here we have the extra summation over all vertices. We note that we have included a conventional prefactor 1 2g here, that is not present in [26,27]. Recall that g = e − v + 1 is the loop order.…”

“…The differential graded complex, called fixed target graph complex, is (OGC n , δ 0 ) where δ 0 is the part of the differential that does not change the number of targets (sinks). It, or rather its isomorphic version with sources instead of targets, is considered in detail in [1].…”

“…In this subsection we are going to define the map Φ : HG S n , d → HOG S n+1 , d . We follow the methods from [27] and [1]. Thanks to Proposition 12 we may consider as well the "skeleton version" HO sk G S n+1 , d HOG S n+1 , d of the oriented graph complex and define our map as…”