Algebraic obstructions and a complete solution of a rational retraction problem

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