Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

Paul Antony Otieno,Vitalis Onyango-Otieno,Jean Baptiste Gatsinzi

doi:10.1155/2020/9385153

Abstract

We consider the complex Grassmannian Grk,n of k-dimensional subspaces of ℂn. There is a natural inclusion in,r:Grk,n↪Grk,n+r. Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in,r in the space of mappings from Grk,n to Grk,n+r for r≥1 and in particular to show that the cohomology of mapGrn,k,Grn,k+r;in,r contains a truncated algebra ℚx/xr+n+k2−nk, where x=2, for k≥2 and n≥4.

Highlights

Introduction e complex GrassmannGr(k, n) is the set of k-planes through the origin in Cn
U(k) × U(n − k) where U(n) is the unitary group ([1], chap. 18). ere is a canonical inclusion in,r: Gr(k, n) ↪ Gr(k, n + r) which is induced by ir: Cn ⟶ Cn+r defined by i(x1, . . . , xn) (x1, . . . , xn, 0, . . . , 0)
E study of the rational homotopy type of function spaces started with om in the case where the codomain is an Eilennerg–Maclane space [2]. e first description of a Sullivan model of function spaces is due to Haefliger [3]

Summary

Introduction

Introduction e complex GrassmannGr(k, n) is the set of k-planes through the origin in Cn. E first description of a Sullivan model of function spaces is due to Haefliger [3]. There is no explicit and complete description of the homotopy type of the component of the inclusion Gr(k, n) ↪ Gr(k, n + r) in the space of mappings from Gr(k, n) to Gr(k, n + r), r ≥ 1 and n ≥ 4.

Results

Conclusion