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Durham e-Theses
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Near-symplectic 2n-manifolds

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
Thesis Date:2014
Copyright:Copyright of this thesis is held by the author
Deposited On:02 Oct 2014 12:33

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