Totally real immersions of surfaces

Abstract

Totally real immersions $f$ of a closed real surface $Σ$ in an almost complex surface $M$ are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes $\frak{M}(f)$ of mappings from $Σ$ into a specific real 5-manifold $E(M)$, while $\frak{M}(f)$ 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 $C^2$, $CP^1\times CP^1$, $CP^2$ and $CP^2 # m\bar{CP^2}$, $m=1,2,...,7$, and all $Σ$ (or, $CP^2 # 8\bar{CP^2}$ and all orientable $Σ$), 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.