# Rational Homotopy Stability for the Spaces of Algebraic Maps

### From NZJM

**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 π_{2}(*X*) a free abelian group of rank *r*. For a based algebraic map , we can assign it a multiple degree under the induced homomorphism . Let be the space of based algebraic maps of degree from into *X*. Under some assumption we prove that the map obtained by composing with induces rational homotopy equivalence up to some dimension, which tends to infinity as the degree grows.

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

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