A relaxed evaluation subgroup

Abstract

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

first version