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\begin{document}



\begin{center}
{\Large \bf 
Rational visibility of a Lie group in the monoid of self-homotopy
equivalences of a homogeneous space 
}
\end{center}

Let $G$ be a connected Lie group and $M$  a homogeneous space admitting 
a left translation by $G$. Let $\text{aut}_1(M)$ denote the identity component
of the monoid of self-homotopy equivalences of $M$. Then the action of
 $G$ on $M$ gives rise to a map $\lambda : G \to \text{aut}_1(M)$. 
The purpose of this article is to investigate 
the injectivity of the homomorphism which $\lambda$ induces on 
the rational homotopy. In particular, {\it the visible degrees} are 
determined explicitly for all the cases of simple Lie groups and their
 associated homogeneous spaces of rank one which are classified by
 Oniscik. 
The main tool for the study is an elaborate rational model 
for an evaluation map, which is constructed in [B-M] and [Ku] 
with the function space model due to Brown and Szczarba [B-S].

\bigskip
\noindent
[B-S]
Brown Jr, E. H. and Szczarba, R. H.: Rational homotopy
type of function
spaces, Trans. Amer. Math. Soc. {\bf 349}(1997), 4931-4951. \\

\bigskip
\noindent
[B-M]U. Buijs and A. Murillo, Basic constructions in rational
	homotopy theory of function spaces, 
Ann. Inst. Fourier (Grenoble) {\bf 56}(2006), 815-838. \\

\bigskip
\noindent
[Ku] K. Kuribayashi, A rational model for the evaluation map, 
Georgian Mathematical Journal {\bf 13}(2006), 127-141.  
\end{document}
