Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form
Let γkn→Gk(Hn) be the Stiefel bundle of quaternionic k-frames in Hn over the the quaternionic Grassmannian Gk(Hn). Let σ denote the first symplectic Pontrjagin form associated with the universal connection on γkn. We show that every 4-form ω on a smooth manifold M can be induced from σ by a smooth map f: M→ Gk(Hn) (for sufficiently large k and n) provided there exists a continuous map f: M→ Gk(Hn) which pulls back the deRham cohomology class of σ (referred as the symplectic Pontrjagin class of γkn) onto that of ω.
Datta, Mahuya, "Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form" (2019). Journal Articles. 830.