Topological morphisms and lifting correspondences of affine algebraic varieties

In general, the algebraic lifting correspondences of projective varieties consider a set of prime ideals with zero genus under morphisms. In this article, I propose the concept and formulation of topological double lifting correspondences of algebraic varieties considering topological based spaces and projective spaces preserving homeomorphism and separable fibrations. The finite sequence of lifting correspondences within a compact projective covering space are introduced. I show that double lifting correspondences admit several homeomorphisms between fundamental groups in the based topological spaces and in the projective covering spaces. The concept of topological endomorphism is extended to the real algebraic varieties in the distinguished Zariski open sets. It is illustrated that the symmetric automorphism in polynomial rings in a closed algebraic field generate equivalent classes of polynomials in subrings along with the formation of fixed points in projective varieties under certain conditions. It is shown that the homeomorphic topological lifting correspondences of projective varieties preserve the genus of algebraic varieties forming the fixed point under the covering functions, which is distinct compared to the Hopf fixed point. I introduce the topological concept of genus deformation morphism of a projective variety and show that there are no homeomorphic lifting correspondences in such cases. However, the covering functions are admissible in such cases considering that the fibers are discrete.