Evaluation subgroups of a map and the rationalized $G$-sequence

Oteng Maphane

doi:10.52737/18291163-2022.14.2-1-10

Abstract

In this paper, we determine, in terms of the Sullivan models, rational evaluation subgroups of the inclusion $ \mathbb{C} P(n)\hookrightarrow \mathbb{C} P(n+k) $ between complex projective spaces and, more generally, the $ G $-sequence of the homotopy monomorphism $ \iota: X\hookrightarrow Y $ between simply connected formal homogeneous spaces for which $ \pi_{\ast}(Y)\otimes \mathbb{Q}$ is finite dimensional.

Highlights

Let us remind the notion of a Gottlieb group
Gottlieb groups play a profound role in topology, covering spaces, fixed point theory, homotopy theory of fibrations, and other fields
Smith and Lupton [4] identified 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

Summary

Introduction

Let us remind the notion of a Gottlieb group (see [3]). Given a based CWcomplex X, an element α ∈ πn(X) is a Gottlieb element of X if (α, idX) : X ∨ Sn → X extends to α : X × Sn → X. Let f : X → Y be a based map of connected finite CWcomplexes. As it was shown in [4], the evaluation at the basepoint of X gives the evaluation map ω : Map(X, Y ; f ) → Y , where Map(X, Y ; f ) is the component of f in the space of mappings from X to Y. Smith and Lupton [4] identified 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. We use a map of complexes of derivations of minimal Sullivan models of mapping spaces to compute rational relative Gottlieb groups of the inclusion CP (n) → CP (n + k). We consider the inclusion ι : X → Y between connected formal homogeneous spaces for which π∗(Y ) ⊗ Q is finite dimensional

Preliminaries

Evaluation subgroups of a map and the Gsequence