A note on the symplectic topology of $b$-manifolds

Frejlich, Pedro; Torres, David Martínez; Miranda, Eva

doi:10.4310/jsg.2017.v15.n3.a4

Cited by 19 publications

(19 citation statements)

References 11 publications

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“…While carrying out this research, the author was made aware that Frejlich, Martinez‐Torres and Miranda were carrying out a project , which overlaps with the results in this paper. Notably, they had independently produced our ‘Construction 1’ and our Theorem .…”

Section: Acknowledgementsmentioning

confidence: 99%

Examples and counter-examples of log-symplectic manifolds

Cavalcanti

2017

Journal of Topology

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We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit bona fide log-symplectic structures and several bona fide log-symplectic manifolds do not admit symplectic structures, for example #mCP 2 #nCP 2 has bona fide log-symplectic structures if and only if m, n > 0 while they only have symplectic structures for m = 1. We introduce surgeries that produce log-symplectic manifolds out of symplectic manifolds and show that any compact oriented log-symplectic four-manifold can be transformed into a collection of symplectic manifolds by reversing these surgeries. Finally we show that if a compact manifold admits an achiral Lefschetz fibration with homologicaly essential fibers, then the manifold admits a log-symplectic structure. Then, using results of Etnyre and Fuller [5], we conclude that if M is a compact, simply connected 4-manifold then M #(S 2 × S 2 ) and M #CP 2 #CP 2 have log-symplectic structures.

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Section: Acknowledgementsmentioning

confidence: 99%

Examples and counter-examples of log-symplectic manifolds

Cavalcanti

2017

Journal of Topology

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“…The following theorem is Theorem B in [11]: Theorem 22. Any cosymplectic manifold of dimension 3 is the singular locus of orientable, closed, bsymplectic manifolds.…”

Section: Claimmentioning

confidence: 99%

Integrable systems on singular symplectic manifolds: From local to global

Cardona¹,

Miranda²

2020

Preprint

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In this article we consider integrable systems on manifolds endowed with singular symplectic structures of order one. By singular symplectic structures of order one we mean structures which are symplectic away from an hypersurface along which the symplectic volume either goes to infinity or to zero in a transversal way (singularity of order one) resulting either in a b-symplectic form or a folded symplectic forms. The hypersurface where the form degenerates is called critical set. In this article we give a new impulse to the investigation of action-angle coordinates for this structures initiated in [30] and [29] by proving an action-angle theorem for folded symplectic integrable systems, establishing new cotangent models for these systems and investigating duality with b-integrable systems via desingularization. We provide global constructions of integrable systems and investigate obstructions for global existence of action-angle coordinates in both scenarios. The new topological obstructions found emanate from the topology of the critical set of the singular symplectic manifold Z. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z.

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“…b-Poisson structures were originally classified in dimension 2 by Radko [1]. Recently, beginning with [2,3], there have been interesting developments in the dynamical and topological aspects of b-Poisson structures in higher dimensions, see [4,5,6,7,8,9,10].…”

Section: Preliminariesmentioning

confidence: 99%