Real Algebraic and Analytic Geometry |

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.

**Full text**, 22p.:
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