Wrinkling h-principles for integral submanifolds of jet spaces

Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding to show that any tangential homotopy can be approximated by a homotopy of topological embeddings with mild singularities. This concept plays an important role in Contact Topology: The loose legendrian h-principle of E. Murphy relies on wrinkled embeddings to manipulate the legendrian front. Similarly, the simplification of legendrian front singularities was proven by D. Álvarez-Gavela by defining the notion of wrinkled legendrian.
This paper and its sequel generalise these ideas to general jet spaces. The main theorem in the present paper proves the analogue of the result by Eliashberg and Mishachev: Any homotopy of the r-order differential information of an embedding can be approximated by a homotopy of embeddings with wrinkle-type singularities (of order r).
The local version of the previous statement, which is of independent interest, says that the holonomic approximation theorem holds over closed manifolds if, instead of sections, we consider multi-valued sections with simple singularities.