Rational homotopy stability for the spaces of rational maps
  Let $\Hol_{x_0}^{{\bf n}} (\C\P^1, X)$ be the space of based holomorphic maps of degree ${\bf n}$ from $\C\P^1$ into a simply connected algebraic variety $X$. Under some condition we prove that the map $\map \Hol_{x_0}^{{\bf n}} (\C\P^1, X). \Hol_{x_0}^{d{\bf n}} (\C\P^1, X).$ obtained by compositing $f \in \Hol_{x_0}^{{\bf n}} (\C\P^1, X)$ with $g(z)=z^d, z \in \C\P^1$ induces rational homotopy equivalence up to some dimension, which tends to infinity as the degree grows. 
Jiayuan Lin