Degrees of maps between Grassmann manifolds

Abstract

Let $f\colon \mathbb{G}_{n,k}\to \mathbb{G}_{m,l}$ be any continuous map between two distinct complex (resp. quaternionic) Grassmann manifolds of the same dimension. We show that the degree of $f$ is zero provided $n,m$ are sufficiently large and $l\geq 2$. If the degree of $f$ is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^{*}$ of every element of a set of algebra generators of $H^{*}(\mathbb{G}_{m,l};\mathbb{Q})$ is determined up to a sign, $\pm$, by the degree of $f$, provided this degree is non-zero.