54 research outputs found
A note on sections of broken Lefschetz fibrations
We show that there exists a non-trivial simplified broken Lefschetz fibration
which has infinitely many homotopy classes of sections. We also construct a
non-trivial simplified broken Lefschetz fibration which has a section with
non-negative square. It is known that no Lefschetz fibration satisfies either
of the above conditions. Smith proved that every Lefschetz fibration has only
finitely many homotopy classes of sections, and Smith and Stipsicz
independently proved that a Lefschetz fibration is trivial if it has a section
with non-negative square. So our results indicate that there are no
generalizations of the above results to broken Lefschetz fibrations. We also
give a necessary and sufficient condition for the total space of a simplified
broken Lefschetz fibration with a section admitting a spin structure, which is
a generalization of Stipsicz's result on Lefschetz fibrations.Comment: 15 pages, 10 figures. We correct the proofs of Theorem 1.2 and
Theorem 1.4 and the argument in Example 5.4. We also modify minor error
Broken Lefschetz fibrations and mapping class groups
The purpose of this note is to explain a combinatorial description of closed
smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of
surface mapping classes, and further explore these connections. This is
obtained via monodromy representations of simplified broken Lefschetz
fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves
that allows us to uniquely determine the isomorphism class of a broken
Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose
monodromies are contained in special subgroups of the mapping class group;
namely, the hyperelliptic mapping class group and in the Torelli group,
respectively, and present various results on them which extend or contrast with
those known to hold for honest Lefschetz fibrations. Lastly, we show that there
are 4-manifolds admitting infinitely many pairwise nonisomorphic relatively
minimal broken Lefschetz fibrations with isotopic regular fibers.Comment: 18 pages, 3 figure
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