Degrees of maps between Grassmann manifolds

Abstract

Let f:G_n,k→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≥2. If the degree of f is ±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^∗(G_m,l;Q) is determined up to a sign, ±, by the degree of f, provided this degree is non-zero.