# Rational Homotopy Stability for the Spaces of Algebraic Maps

### New Zealand Journal of Mathematics

Vol. 38, (2008), Pages 179-186

J. Lin

Department of Mathematics

SUNY Canton

34 Cornell Drive

Canton, NY 13617

U.S.A

Abstract Let X be a path connected nilpotent (e.g. simply connected) complex algebraic variety with π2(X) a free abelian group of rank r. For a based algebraic map $f: ({\mathbb C} {\mathbb P}^1, \infty) \rightarrow (X, x_0)$, we can assign it a multiple degree ${\mathbf n}=f_*(1)$ under the induced homomorphism $f_*: \pi_2({\mathbb C} {\mathbb P}^1) \rightarrow \pi_2(X)$. Let ${\rm Alg}_{x_0}^{\mathbf n} ({\mathbb C} {\mathbb P}^1, X)$ be the space of based algebraic maps of degree $\mathbf n$ from ${\mathbb C}{\mathbb P}^1$ into X. Under some assumption we prove that the map ${\rm Alg}_{x_0}^{\mathbf n} ({\mathbb C} {\mathbb P}^1, X) \to {\rm Alg}_{x_0}^{d\mathbf n} ({\mathbb C} {\mathbb P}^1, X)$ obtained by composing $f \in {\rm Alg}_{x_0}^{\mathbf n} ({\mathbb C} {\mathbb P}^1, X)$ with $g(z)=z^d, z \in {\mathbb C} {\mathbb P}^1$ induces rational homotopy equivalence up to some dimension, which tends to infinity as the degree ${\mathbf n}$ grows.

Keywords Sullivan-Haefliger model, rational homotopy equivalence, stability property, nilpotent space.

Classification (MSC2000) 55P62, 14F45, 14E99.