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Register a new userCommutators, Lefschetz fibrations and the signatures of surface bundles
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We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.
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We examine the action of the fundamental group $Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $widetilde{mathcal B}$ on $m$ strings on a suitable space of $mtimes m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${mathcal S}_c:widetilde{mathcal B}to GL_m({mathbb{Z}}[Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.
In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $pi$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $pi$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually overtwisted tight contact structure on the negative parabolic torus bundle with monodromy $-T^n$ ($n<0$), we prove that it is Stein fillable. By the way, we give a partial answer to a conjecture of Golla and Lisca.
We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.
We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebruch. The 3-manifolds are related to a Lefschetz fibration of the complex hyperbolic orbifold.
Periodic surface homemorphisms (diffeomorphisms) play a significant role in the the Nielsen-Thurston classification of surface homeomorphisms. Periodic surface homeomorphisms can be described (up to conjugacy) by using data sets which are combinatorial objects. In this article, we start by associating a rational open book to a slight modification of a given data set, called marked data set. It is known that every rational open book supports a contact structure. Thus, we can associate a contact structure to a periodic map and study the properties of it in terms combinatorial conditions on marked data sets. In particular, we prove that a class of data sets, satisfying easy-to-check combinatorial hypothesis, gives rise to Stein fillable contact structures. In addition to the above, we prove an analogue of Moris construction of explicit symplectic filling for rational open books. We also prove a sufficient condition for Stein fillability of rational open books analogous to the positivity of monodromy in honest open books as in the result of Giroux and Loi-Piergallini.
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