Abstract
A 3-form on 7 dimensional vector space is called multisymplectic if it satisfies some natural non-degeneracy requirement. It is well known that there are 8 orbits (or types) of multisymplectic 3-forms on under the canonical action of the general linear group and that two types are open.
This leads to 8 types of global multisymplectic 3-forms on 7-dimensional manifolds without boundary. The existence of a global multisymplectic 3-form of a fixed type is a classical problem in differential topology which is equivalent to the existence of a certain G-structure.
The open types are the most interesting cases as they are equivalent to G_2-structure. The existence of these two structures is a well known and solved problem.
In this article is solved (under some convenient assumptions) the problem of the existence of multisymplectic 3-forms of the remaining types