Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form

Abstract

Let $$\gamma ^n_k\rightarrow G_k(\mathbb {H}^n)$$ be the Stiefel bundle of quaternionic k-frames in $$\mathbb {H}^n$$ over the the quaternionic Grassmannian $$G_k(\mathbb {H}^n)$$ . Let $$\sigma $$ denote the first symplectic Pontrjagin form associated with the universal connection on $$\gamma ^n_k$$ . We show that every 4-form $$\omega $$ on a smooth manifold M can be induced from $$\sigma $$ by a smooth map $$f:M\rightarrow G_k(\mathbb {H}^n)$$ (for sufficiently large k and n) provided there exists a continuous map $$f_0:M\rightarrow G_k(\mathbb {H}^n)$$ which pulls back the deRham cohomology class of $$\sigma $$ (referred as the symplectic Pontrjagin class of $$\gamma _k^n$$ ) onto that of $$\omega $$ .