Second Order Homological Obstructions and Global Sullivan–type Conditions on Real Algebraic Varieties

It is well-known that the existence of non–algebraic Z/2–homology classes of a real algebraic manifold Y is equivalent to the existence of non–algebraic elements of the unoriented bordism group of Y and generates (first order) obstructions which prevent the possibility of realizing algebraic properties of smooth objects defined on Y . The main aim of this paper is to investigate the existence of smooth maps f : X −→ Y between a real algebraic manifold and Y not homotopic to any regular map when Y has totally algebraic homology, i.e, when the first order obstructions on Y do not occur. In this situation, we also discover that the homology of Y generates obstructions: the second order obstructions on Y . In particular, our results establish a clear distinction between the property of a smooth map f to be bordant to a regular map and the property of f to be homotopic to a regular map. As a byproduct, we obtain two global versions of Sullivan’s condition on the local Euler characteristic of a real algebraic set and give obstructions to the existence of algebraic tubular neighborhoods of algebraic submanifolds of Rn.