Narrow Results

A surface bundle determines a monodromy representation recording the twisting of the fiber under transport around a closed path in the base space. The fascinating relation between these monodromy representations and the Thurston classification of surface automorphisms will be studied. In this note we deal with a simple and interesting case: the fibrations of Fadell and Neuwirth.

We show that totally real elliptic Lefschetz fibrations admitting a real section are classified by their "real loci" which can be encoded in terms of a combinatorial object that we call a necklace diagram. By means of necklace diagrams, we obtain an explicit list of certain classes of totally real elliptic Lefschetz fibrations.

Consider an arrangement ${𝒜}$ of homogeneous hyperplanes in ${ℂ}^n,$ with complement ${ℳ}({𝒜})$. The (co)homology of ${ℳ}({𝒜})$ with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto ${ℂ}^{\ast },$ endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of ${\pi }_1({ℳ}({𝒜}))$ by the commutator of its length zero subgroup, which didn’t appear in the literature before.

In Sec. 2, we state a conjecture of $a$-monodromicity for the first homology, which is of a different nature with respect to the known results. Let $\mathrm{\Gamma }$ be the graph of double points of ${𝒜}:$ we conjecture that if $\mathrm{\Gamma }$ is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of ${𝒜}$. We show the truth of the conjecture under some stronger hypotheses.