Relative Gottlieb Groups of Embeddings between Complex Grassmannians

J B Gatsinzi,Luca Vitagliano

doi:10.1155/2021/1417769

Abstract

Let Gr k , n be the complex Grassmann manifold of k -linear subspaces in ℂ n . We compute rational relative Gottlieb groups of the embedding i : Gr k , n ⟶ Gr k , n + r and show that the G -sequence is exact if r ≥ k n − k .

Highlights

In [2], Lee and Woo introduce relative evaluation groups Grnel(Y, X; f) and obtain a long sequence,· · · ⟶ Grne+l1(Y, X; f) ⟶ Gn(X) ⟶ Gn(Y, X; f) (1) ⟶ Grnel(Y, X; f) ⟶ · · ·, called G-sequence [5]. is sequence is exact in some cases, for instance, if f is a homotopy monomorphism [6].2
Let f: X ⟶ Y be a map between connected CW complexes, where X is of finite type and φ: (∧V, d) ⟶ (∧W, d) its Sullivan model. e long exact sequence induced by the map f∗: map(X, X; 1X) ⟶ map(X, Y; f) on rational homotopy groups is equivalent to the long exact sequence of φ∗: Der(∧W, d) ⟶ Der(∧V, ∧W; φ)
As p is a quasiisomorphism, the G-sequence of the inclusion is computed from the long exact sequence induced by the cone of the map: φ∗: Der ∧W, H∗(∧W); p􏼁 ⟶ Der ∧V, H∗(∧W); φ􏼁

Summary

Introduction

A connected topological space X is called formal if there exists a quasi-isomorphism (∧V, d) ⟶ H∗(X, Q), where (∧V, d) is a Sullivan model of X. E inclusion Gr(2, 4) ⟶ Gr(2, 7) has a Sullivan model: φ: ∧ x2, x4, x11, x13􏼁, d􏼁 ⟶ ∧ y2, y4, y5, y7􏼁, d􏼁, (10) If X is connected and (∧V, d) is the minimal Sullivan model of X,

Results

Conclusion