Derivations of a Sullivan Model and the Rationalized  G -Sequence

Oteng Maphane

doi:10.1155/2021/6687527

Abstract

Let G k , n ℂ for 2 ≤ k < n denote the Grassmann manifold of k -dimensional vector subspaces of ℂ n . In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the G -sequence of the inclusion G 2 , n ℂ ↣ G 2 , n + r ℂ for r ≥ 1 .

Highlights

In [3], Smith and Lupton identify the homomorphism induced on rational homotopy groups by the evaluation map ω: (Map(X, Y; f)) ⟶ Y, in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of f
In [5], the authors use a map of complexes of derivations of minimal Sullivan models of mapping spaces to compute rational relative Gottlieb groups of the inclusion G2,n(C)↣G2,n+1(C) between complex Grassmannians
All vector spaces and algebras are taken over a field Q of rational numbers

Summary

Sn idX h

X, where h: Sn ⟶ X is a representative of a and ▽ is the folding map. Let f: X ⟶ Y be a based map of connected finite CW-complexes. In [3], Smith and Lupton identify the homomorphism induced on rational homotopy groups by the evaluation map ω: (Map(X, Y; f)) ⟶ Y, in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of f. In [5], the authors use a map of complexes of derivations of minimal Sullivan models of mapping spaces to compute rational relative Gottlieb groups of the inclusion G2,n(C)↣G2,n+1(C) between complex Grassmannians. We generalize their work to compute rational relative Gottlieb groups of the inclusion Gk,n(C)↣Gk,n+r (C), r ≥ 1 between complex Grassmannians for 2 ≤ k < n

Preliminaries

Derivations of a Sullivan Model and the GSequence