Totally real immersions of surfaces

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 $\frak{M}(f)$ of mappings from $\Sigma$ 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 $\Sigma$ (or, CP^2 # 8\bar{CP^2} 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.Comment: 71 page

The ?2-cohomology of hyperplane complements

We compute the l^2-Betti numbers of the complement of any finite collection of affine hyperplanes in complex n-space. At most one of the l^2-Betti numbers is non-zero. <br/

Commensurability of graph products

Abstract We de ne graph products of families of pairs of groups and study the question when two such graph products are commensurable. As an application we prove linearity of certain graph products

Odd-dimensional Charney-Davis conjecture

More than once we have heard that the Charney-Davis Conjecture makes sense only for odd-dimensional spheres. This is to point out that in fact it is also a statement about even-dimensional spheres.Comment: 3 pages, no figure