Real Algebraic and Analytic Geometry
Submission: 2003, October 28.
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 possibili\-ty of realizing algebraic properties of smooth objects defined on $Y$. The main aim of this paper is to investigate the existence of smooth maps $f:X \lra Y$ 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 $\R^n$.
Keywords and Phrases: Real algebraic homotopy classes, Second order obstructions, Sullivan-type conditions, Real algebraic manifolds, Real algebraic sets.
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