ABSTRACT. We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector vanishes on the singularities, where we give its local form explicitly.

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 ω n−1 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 nearsymplectic 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. arXiv:1211.5859v3 [math.SG]

We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.This section follows notations used in [20] and [21], where Bott-Morse foliations on dimension 3 were described.Let M m be a closed, orientable, smooth manifold of dimension m, for m ≥ 3. Let F be a codimension-one smooth foliation with singularities on M . Denote by Sing(F) the set of singular points of F.

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