Algebraic obstructions and a complete solution of a rational retraction problem

Abstract

Abstract. For each compact smooth manifold W containing at least two points we prove the existence of a compact nonsingular algebraic set Z and a smooth map g : Z −→ W such that, for every rational diffeomorphism r : Z −→ Z and for every diffeomorphism s : W −→ W where Z and W are compact nonsingular algebraic sets, we may fix a neighborhood U ofwhich does not contain any regular rational map. Furthermore s −1 • g • r is not homotopic to any regular rational map. Bearing in mind the case in which W is a compact nonsingular algebraic set with totally algebraic homology, the previous result establishes a clear distinction between the property of a smooth map f to represent an algebraic unoriented bordism class and the property of f to be homotopic to a regular rational map. Furthermore we have: every compact Nash submanifold of R n containing at least two points has not any tubular neighborhood with rational retraction.