Maps from Grassmannians of 2-planes to projective spaces

TL;DR

This paper constructs specific maps from Grassmannians of 2-planes to projective spaces using quaternions and octonions, revealing topological properties and estimating the Lusternik-Schnirelmann category.

Contribution

It introduces new quaternionic and octonionic maps from Grassmannians to projective spaces, analyzing their topological features and applications.

Findings

All maps induce isomorphisms on fundamental groups.

Two maps are submersions.

New estimates for Lusternik-Schnirelmann category of Grassmannians.

Abstract

Using quaternions and octonions, we construct some maps from the Grassmannian of 2-dimensional planes of $\mathbb{R}^n$, $\mathrm{Gr}_2(\mathbb{R}^n)$, to the projective space $\mathbb{R}\mathrm{P}^k$, for certain values of $n$ and $k$. All of our maps induce an isomorphism at the level of fundamental groups, and two of them are shown to be submersions. As an application, we obtain new estimates of the Lusternik-Schnirelmann category of $\mathrm{Gr}_2(\mathbb{R}^n)$ for specific values of $n$.