Multi-directed graph complexes and quasi-isomorphisms between them II: Sourced graphs

TL;DR

This paper proves that certain sourced graph complexes are quasi-isomorphic to standard Kontsevich graph complexes, revealing their homological equivalence and potential applications in deformation theory and topology.

Contribution

It establishes quasi-isomorphisms between sourced and standard graph complexes, extending to multi-directed complexes and developing a theory of graph complexes with arbitrary edge types.

Findings

Oriented graph complex inclusion is a quasi-isomorphism.

Homology of sourced graph complex equals that of Kontsevich's graph complex.

All multi-directed graph complexes are quasi-isomorphic.

Abstract

We prove that the inclusion from oriented graph complex into graph complex with at least one source 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 $T_{\rm poly}^{\geq 1}$ of degree at least one, and to the hairy graph complex which 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.