We show how the machine invented by S. Merkulov [19, 20, 22] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natural operators acting on vector fields and connections.

Abstract. In the paper we prove that the primitive part of the Sinha homology spectral sequence E 2-term for the space of long knots is rationally isomorphic to the homotopy E 2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces. 1.

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The homology of Kontsevich’s commutative graph complex parametrizes finite type invariants of odd-dimensional manifolds. This graph homology is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give two different proofs (one algebraic, one geometric) that the commutative graph complex is quasi-isomorphic to the quotient complex obtained by modding out by graphs with cut vertices. This quotient complex has the advantage of being smaller and hence more practical for computations. In addition, it supports a Lie bialgebra structure coming from a bracket and cobracket we defined in a previous paper. As an application, we compute the rational homology groups of the commutative graph complex up to rank 7. 1.