On hidden face contributions of configuration space integrals for long embeddings

Configuration space integrals are powerful tools for studying the homotopy type of the space of long embeddings in terms of a combinatorial object called a graph complex. It is unknown whether these integrals give a cochain map due to potential obstructions called hidden faces. The purpose of this paper is to address these hidden faces by modifying configuration space integrals: we incorporate the acyclic bar complex of some dg algebra into the original graph complex, without changing its cohomology. Then, we give a cochain map from the new graph complex to the de Rham complex of the space of long embeddings modulo immersions, by combining the original configuration space integrals with Chen's iterated integrals. As the original complex, we choose a modified graph complex so that it is quasi‐isomorphic to both the hairy graph complex and a graph complex introduced in the context of embedding calculus.