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Homology of real algebraic varieties and morphisms to spheres

Öztürk, Ali
Let X and Y be affine nonsingular real algebraic varieties. One of the classical problems in real algebraic geometry is whether a given C1 mapping f : X ! Y can be approximated by regular mappings in the space of C1 mappings. In this thesis, we obtain some sufficient conditions in the case when Y is the standard sphere Sn. In the second part of the thesis, we study mainly the kernel of the induced map on homology i : Hk(X,R) ! Hk(XC,R), where i : X ! XC is a nonsingular projective complexification. First, using Lefshcetz Hyperplane Section Theorem we study KHk(X \ H,R), where H is a hyperplane. In the remaining part, we relate KHk(X,R) to the realization of cohomology classes of XC by harmonic forms.