Rational surfaces and regular maps into the 2-dimensional sphere

Abstract.

Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a ${\cal C}^\infty$ mapping, $f:X\rightarrow Y$, can be approximated by regular mappings in the space of ${\cal C}^\infty$ mappings, ${\cal C}^\infty(X,Y)$, equipped with the ${\cal C}^\infty$ topology. In this paper, we obtain some results concerning this problem when the target space is the 2-dimensional standard sphere and X has a complexification $X_{\mathbb C}$ that is a rational (complex) surface. To get the results we study the subgroup $H^2_{{\mathbb C}-\text{alg}}(X,{\mathbb Z})$ of the second cohomology group of X with integer coefficients that consists of the cohomology classes that are pullbacks, via the inclusion mapping $X\rightarrow X_{\mathbb C}$, of the cohomology classes in $H^2(X_{\mathbb C},{\mathbb Z})$ represented by complex algebraic hypersurfaces.