Algebraic obstructions and a complete solution of a rational retraction problem

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 diffeomorphis s : W' --> W where Z' and W are compact nonsingular algebraic sets, we may fix a neighborhood U of s(-1) o g o r in C-infinity (Z', W') which does not contain any regular rational map. Furthermore s(-1) o g o 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.