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\begin{document}$$\mathcal {E}(X)$$\end{document} be the group of homotopy classes of self homotopy equivalences for a connected CW complex X. We consider two classes of maps, E\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}$$\end{document}-maps and co-E\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}$$\end{document}-maps. They are defined as the maps X→Y\documentclass[12pt]{minimal}
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\begin{document}$$X\rightarrow Y$$\end{document} that induce homomorphisms E(X)→E(Y)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}(X)\rightarrow \mathcal {E}( Y)$$\end{document} and E(Y)→E(X)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}(Y)\rightarrow \mathcal {E}(X)$$\end{document}, respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an E\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}$$\end{document}-equivalence relation between rationalized spaces XQ\documentclass[12pt]{minimal}
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\begin{document}$$X_{{\mathbb Q}}$$\end{document} and YQ\documentclass[12pt]{minimal}
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\begin{document}$$Y_{{\mathbb Q}}$$\end{document} as a geometric realization of an isomorphism E(XQ)≅E(YQ)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})$$\end{document}.
