The framework of this study is a closed manifold of dimension at least six that is provided with a nonzero De Rham cohomology class. The aim is to create tools to address the next problem: two closed non-singular (without zeroes) 1-forms in the fixed class are always isotopic? The general answer to the question is no, and a K-theoretical obstruction is expected. It is always possible to connect two non-singular closed 1-forms by a path that remains in the cohomology class; the isotopy of the two ends of the path is equivalent to find a relative homotopy of the path to another one made of non-singular 1-forms only. We introduce two kinds of pseudo-gradients for each positive number L: those with an L-elementary link and those that we call L-transverse. They form a class of vector fields adapted to the 1-forms that allows to do an algebraic reading associated with the path. This reading is similar to that made in the theory of Hatcher-Wagoner who treated the isotopy problem of real-valued functions without critical points. We manage to find L, a number large enough to deform a path of 1-forms with only two critical indices into another one with an L-transverse equipment in normal form. The zeroes of such a path that are born together, die together and moreover, the associated Cerf-Novikov graphic is closed : the cited algebraic reading belongs to some K_2, which is the starting point for the definition of an obstruction for two non-singular closed 1-forms to be isotopic.
