Hairy graphs to ribbon graphs via a fixed source graph complex

2019/12/19 by Assar Andersson, Andersson, Assar, Marko Živković +1 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1912.09438

openalex publication_date 2019/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (OGC₁,d₀) 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 GC²_n. This gives a new connection between the standard and the oriented version of Kontsevich's graph complex.

Cited by

• Getzler-Kapranov complexes and moduli stacks of curves
• Oriented hairy graphs and moduli spaces of curves

Related

• Oriented hairy graphs and moduli spaces of curves
• Ribbon graphs and the fundamental group of surfaces
• Multi-directed graph complexes and quasi-isomorphisms between them II: Sourced graphs
• Semistable Graph Homology
• A dual point of view on the ribbon graph decomposition of moduli space