A relaxed evaluation subgroup

TL;DR

This paper introduces the relaxed evaluation subgroup in homotopy theory, generalizing the evaluation subgroup, and explores its properties using rational homotopy theory and Sullivan models.

Contribution

It defines the relaxed evaluation subgroup and analyzes its structure, especially in rational homotopy theory, providing new insights and comparisons with existing subgroups.

Findings

The relaxed evaluation subgroup equals the rational homotopy group when the map induces an injection.

Uses Sullivan models to analyze the subgroup in rational homotopy theory.

Provides examples comparing the relaxed subgroup with other subgroups.

Abstract

Let $f:X\to Y$ be a pointed map between connected CW-complexes. As a generalization of the evaluation subgroup $G_*(Y,X;f)$, we will define the {\it relaxed evaluation subgroup} ${\mathcal G}_*(Y,X;f)$ in the homotopy group $\pi_*(Y)$ of $Y$, which is identified with ${\rm Im} \pi_*(\tilde{ev})$ for the evaluation map $\tilde{ev} :map(X,Y;f)\times X\to Y$ given by $\tilde{ev} (h,x)=h(x)$. Especially we see by using Sullivan model in rational homotopy theory for the rationalized map $f_{\Q}$ that ${\mathcal G}_*(Y_{\Q},X_{\Q};f_{\Q})=\pi_*(Y)\otimes \Q$ if the map $f$ induces an injection of rational homotopy groups. Also we compare it with more relaxed subgroups by several rationalized examples.