Rational homotopy of maps between certain complex Grassmann manifolds

Abstract

Let $G_{n,k}$ denote the complex Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb{C}^n$. Assume $l,k\le \lfloor n/2\rfloor$. We show that, for sufficiently large $n$, any continuous map $h:G_{n,l}\to G_{n,k}$ is rationally null homotopic if $(i)~ 1\le k< l,$ $(ii)~2<l<k< 2(l-1)$, $(iii)~1<l<k$, $l$ divides $n$ but $l$ does not divide $k$.