Immersions in a Quaternionic Grassmannian inducing a given 4-form

TL;DR

The paper demonstrates that any 4-form on a smooth manifold can be realized via immersions into quaternionic Grassmannians, linking differential forms to geometric structures through universal connections.

Contribution

It establishes conditions under which a 4-form can be induced from a universal symplectic Pontrjagin form via smooth immersions into quaternionic Grassmannians, extending geometric realization techniques.

Findings

Any 4-form can be induced from the universal form via immersion.

Existence of a continuous map pulling back cohomology classes is sufficient.

Results hold for sufficiently large dimensions of Grassmannians.

Abstract

Let $Gr_k(\H^n)$ be the Grassmannian manifold of Quaternionic $k$-planes in $\H^n$ and let $\gamma^n_k\to Gr_k(\H^n)$ denote the Stiefel bundle of quaternionic $k$-frames in $\H^n$. 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 immersion $f:M\to Gr_k(\H^n)$ (for sufficiently large $k$ and $n$) provided there exists a continuous map $f_0:M\to Gr_k(\H^n)$ which pulls back the cohomology class of $\sigma$ onto that of $\omega$.