Rational homotopy of maps between certain complex Grassmann manifolds

TL;DR

This paper investigates the rational homotopy classes of maps between complex Grassmann manifolds, establishing conditions under which such maps are rationally null homotopic for large dimensions.

Contribution

It provides new criteria determining when continuous maps between certain complex Grassmannians are rationally null homotopic, based on divisibility and dimension constraints.

Findings

Maps are rationally null homotopic under specified conditions.

Results depend on divisibility of dimensions and relative sizes of k and l.

Findings hold for sufficiently large n.

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 $k$.