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Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding toshow that any tangential homotopy can be approximated by a homotopy oftopological embeddings with mild singularities. This concept plays an importantrole in Contact Topology: The loose legendrian h-principle of E. Murphy relieson wrinkled embeddings to manipulate the legendrian front. Similarly, thesimplification of legendrian front singularities was proven by D.\'Alvarez-Gavela by defining the notion of wrinkled legendrian.  This paper and its sequel generalise these ideas to general jet spaces. Themain theorem in the present paper proves the analogue of the result byEliashberg and Mishachev: Any homotopy of the r-order differential informationof an embedding can be approximated by a homotopy of embeddings withwrinkle-type singularities (of order r).  The local version of the previous statement, which is of independentinterest, says that the holonomic approximation theorem holds over closedmanifolds if, instead of sections, we consider multi-valued sections withsimple singularities.

Wrinkling h-principles for integral submanifolds of jet spaces