Nijenhuis infinity and contractible dg manifolds

Abstract

We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar construction on the quadratic operad of homologically trivial dg Lie algebras. As a by product we obtain a strong homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifold.