# Multi-directed graph complexes and quasi-isomorphisms between them I: oriented graphs

## TL;DR

This paper establishes a direct quasi-isomorphism between Kontsevich's graph complex and the oriented graph complex, extending the result to a sequence of multi-oriented graph complexes, which are important in deformation theory.

## Contribution

It provides an explicit quasi-isomorphism between two key graph complexes and extends this to a sequence of multi-oriented complexes, offering new insights in deformation theory.

## Key findings

- Constructed a direct quasi-isomorphism between GC_n and OGC_{n+1}
- Extended the quasi-isomorphism to a sequence of multi-oriented graph complexes
- Confirmed the complexes' role in deformation theory of multi-oriented props

## Abstract

We construct a direct quasi-isomorphism from Kontsevich's graph complex GC_n to the oriented graph complex OGC_{n+1}, thus providing an alternative proof that the two complexes are quasi-isomorphic. Moreover, the result is extended to the sequence of multi-oriented graph complexes, where GC_n and OGC_{n+1} are the first two members. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov.