Knowledge Bank

Totally real immersions f of a closed real surface Sigma in an almost complex surface M are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes M(f) of mappings from Sigma into a specific real 5-manifold E(M), while M(f) themselves are subject to a single cohomology constraint. This follows from Gromov's observation that totally real immersions satisfy the h-principle. For the receiving complex surfaces C-2, CP1 x CP1, CP2 and CP2#mCP(2), m = 1, 2,...,7, and all Sigma (or, Cp-2 #8CP and all orientable Sigma), we illustrate the above nonconstructive result with explicit examples of immersions realizing all possible equivalence classes. We also determine which equivalence classes contain totally real embeddings, and provide examples of such embeddings for all classes that contain them.