Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

Abstract

We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the homology of standard Kontsevich’s graph complex. This result may have applications in theory of multi-vector fields $T_{\textrm{poly}}^{\geq 1}$ of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types.

Hairy graphs to ribbon graphs via a fixed source graph complex

“…2-valent vertices with one incoming and one outgoing edge. This condition will not change the homology, as shown in [24,Subsection 3.2].…”

“…Directed, oriented and sourced graph complexes. The directed, oriented and sourced graph complexes DGC n , OGC n and SGC n are defined in [24]. In this paper, we will only consider single directional complexes.…”

“…Let (SGC n , δ) be the graph complex consisting of directed graphs that contain at least one source vertex, with the edge contracting differential δ. In [24], the second author showed that the projection (SGC n , δ) → (OGC n , δ) is a quasi-isomorphism.…”

“…However, as there is an isomorphism Inv : OGC → OGC that inverts all the edges, thus mapping source vertices to target vertices and vice versa, we may shift freely between considering source vertices and target vertices. To keep notation consistent with what is previously established in [24], we shall mainly consider the differential that preserves the number of source vertices, which we will also denote by d 0 .…”

Quantizations of Lie bialgebras, duality involution and oriented graph complexes

“…In [ Ž2,Subsection 3.4] the isomorphic skeleton version of oriented graph complex O sk G d is introduced. It is the complex of essentially the same graphs where vertices that are at least 3-valent are seen as skeleton vertices, and the structure of edges and 2-valent vertices between them is seen as a skeleton edge.…”