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

It is well-know that the existence of non-algebraic Z/2homology 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 estabilish 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 abstructions to the existence of algebraic tabular neighborhoods of algebraic submanifolds of Rn.