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

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≥1poly 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.