Real Algebraic and Analytic Geometry
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65. Riccardo Ghiloni:
Second Order Homological Obstructions and Global Sullivan-type Conditions on Real Algebraic Varieties.

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Submission: 2003, October 28.

Abstract:
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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