Degrees of maps between Grassmann manifolds

TL;DR

This paper investigates the degrees of continuous maps between complex Grassmann manifolds, establishing conditions under which the degree is zero or one, and characterizing maps with degree ±1.

Contribution

It provides new results on the degree of maps between Grassmann manifolds, including conditions for zero degree, and characterizes maps with degree ±1, extending to quaternionic cases.

Findings

Degree of maps is zero for large dimensions unless they are homotopy equivalences.

Maps with degree ±1 are homotopy equivalences when the dimensions match.

The induced map on cohomology is determined up to sign if the degree is non-zero.

Abstract

Let $f:G_{n,k}\longrightarrow G_{m,l}$ be any continuous map between any two distinct complex Grassmann manifolds of the same dimension where the target is not the complex projective space. We show that, for any given $k,l$, the degree of $f$ is zero provided that $m,n$ are sufficiently large. 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 elements of a set of algebra generators of $H^*(G_{m,l};\mathbb{Q})$ is determined upto a sign, $\pm$, if the degree of $f$ is non-zero. Our proofs cover the case of quaternionic Grassmann manifolds as well.