Andrzej Derdzinski
Ohio State University
Totally real immersions of surfaces (joint work with T. Januszkewicz)
Abstract:
We study totally real immersions of closed real surfaces S in an arbitrary almost complex surface M (that is, a real four-manifold with an almost complex structure). Using the fact that these immersions satisfy Gromov's h-principle, we show that, for any given S and M, the connected components of the set of such immersions are in a bijective correspondence with the set of those homotopy classes of mappings from S into a suitable five-dimensional manifold E(M) which obey a certain cohomology condition. (Specifically, E(M) is the unit circle bundle of the tensor square of det TM.) In particular, a totally real immersion of S in M exists if and only if some mapping from S into E(M) satisfies the cohomology condition just mentioned. This result seems to provide a complete description of totally real immersions of any given S in a given M. Yet, being based on h-principle, it offers no clue as to what such an immersion, representing a specific connected component, might look like. This is why we provide explicit (or nearly explicit) examples of totally real immersions and embeddings of closed real surfaces S in a few familiar complex surfaces M (namely, the complex plane C^2, the product CP^1 x CP^1, as well the complex projective plane CP^2 and manifolds obtained from it by blowing up fewer than nine points). For each of those M, and every S, we settle the existence question for totally real immersions directly (i.e., without invoking the h-principle), and