Suspending the Cartan Embedding of $${{\mathbb H}P^n}$$ Through Spindles and Generators of Homotopy Groups

Abstract

We provide an equivariant suspension of the Cartan embedding of the symmetric space $${S^{4n+3} \to \mathbb {H}P^n \hookrightarrow Sp(n+1)}$$ ; this construction furnishes geometric generators of the homotopy group of π 4n+6 Sp(n + 1). We study the topology and geometry of the image of this generator; in particular we show that it is a spindle, minimal with respect to the biinvariant metric from Sp(n + 1). This spindle also admits a different metric of positive curvature away from the cone singular point.