Rational homotopy of maps between certain complex Grassmann manifolds

Abstract

Abstract Let G n,k denote the complex Grassmann manifold of k-dimensional vector subspaces of ℂ n . Assume l,k ≤ ⌊ n/2⌋. We show that, for sufficiently large n, any continuous map h : G n,l → G n,k is rationally null homotopic if (i) 1 ≤ k < l, (ii) 2 < l < k < 2(l − 1), (iii) 1 < l < k, l divides n but l does not divide k.