Rational homotopy of mapping spaces between complex Grassmannians

Abstract

The complex Grassmann Gr(k, n) is the space of k dimensional subspaces of ℂ ⁿ . It is a complex manifold of complex dimension k(n − k). There is a natural inclusion i_{k ,n} : Gr(k, n) ↪ Gr(k, n + r). In this paper, we use Sullivan models to compute the rational homotopy type of the component of the inclusion Gr(2, n) ↪ Gr(2, n + r) in the space of mappings from Gr(2, n) to Gr(2, n + r), r ≥ 1. We show in particular that map(Gr(2, n), Gr(2, n + 1); i_n ) has the rational homotopy type of a product of odd spheres.