Maps from Grassmannians of 2-planes to projective spaces

TL;DR

The paper constructs explicit maps $\mathrm{Gr}_2(\mathbb{R}^n)\to \mathbb{R}\mathrm{P}^k$ that induce $\pi_1$-isomorphisms by leveraging quaternionic and octonionic structures, notably defining $\nu_4$ for $n=4$ and $\nu_8$ for $n=8$, and proving they are submersions via equivariant methods. It extends these constructions to higher dimensions (e.g., $\nu_{16}$) through block decompositions and discusses restrictions and extensions, while using Spin$(8)$ triality and related symmetry to establish equivariance. The authors apply these maps to obtain sharper Lusternik-Schnirelmann category estimates for $\mathrm{Gr}_2(\mathbb{R}^n)$, including exact values $\mathrm{cat}(\mathrm{Gr}_2(\mathbb{R}^7))=8$ and $\mathrm{cat}(\mathrm{Gr}_2(\mathbb{R}^8))=9$, and provide a framework for improved bounds via classifying maps to $\mathbb{R}\mathrm{P}^k$. Overall, the work links explicit geometric constructions with topological invariants of Grassmannians, offering new tools for computing LS-categories through equivariant and fibration structures.

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 $n$.

Theorems & Definitions (35)

• Theorem 1.1: KS91
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof : Proof of Proposition \ref{['prop:homeom-hq']}