Algebraic obstructions and a complete solution of a rational retraction problem

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 of s -1 o g o r in C∞(Z', W') which does not contain any regular rational map. Furthermore s -1 o g or 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.