Fake real planes: exotic affine algebraic models of R-2

We study real rational models of the euclidean plane R-2 up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane RP2 is well known: up to birational diffeomorphisms, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to R-2 but which is not isomorphic to the real affine plane. We prove that fake planes exist by giving many examples and we tackle the question: do there exist fake planes whose real locus is not birationally diffeomorphic to the real affine plane?