VERA-SANCHEZ, RAMON (2014) Near-symplectic 2n-manifolds. Doctoral thesis, Durham University.
We give a generalization of the concept of near-symplectic structures to 2n dimensions. According to our definition, a closed 2-form on a 2n-manifold M is near-symplectic, if it is symplectic outside a submanifold Z of codimension 3, where the (n-1)-th power of the 2-form vanishes. We depict how this notion relates to near-symplectic 4-manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration, or BLF, as a singular map with indefinite folds and Lefschetz-type singularities. We show that given such a map on a 2n-manifold over a symplectic base of codimension 2, then the total space carries such a near-symplectic structure, whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension--3 singular locus Z . We describe a splitting property of the normal bundle N_Z that is also present in dimension four. A tubular neighbourhood for Z is provided, which has as a corollary a Darboux-type theorem for near-symplectic forms.
|Item Type:||Thesis (Doctoral)|
|Award:||Doctor of Philosophy|
|Keywords:||Symplectic geometry, broken Lefschetz fibrations, singularities of smooth mappings, fold, singular differential 2-forms, contact topology|
|Faculty and Department:||Faculty of Science > Mathematical Sciences, Department of|
|Copyright:||Copyright of this thesis is held by the author|
|Deposited On:||02 Oct 2014 12:33|