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The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin, Sheila Sandon (2015)

Journal of the European Mathematical Society

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on ℝ 2 n × S 1 and ℝ P 2 n + 1 . On the other hand we also show by elementary arguments that the...

The geography of simply-connected symplectic manifolds

Mi Sung Cho, Yong Seung Cho (2003)

Czechoslovak Mathematical Journal

By using the Seiberg-Witten invariant we show that the region under the Noether line in the lattice domain ℤ × ℤ is covered by minimal, simply connected, symplectic 4-manifolds.

Toric structures on near-symplectic 4-manifolds

David T. Gay, Margaret Symington (2009)

Journal of the European Mathematical Society

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of...

Twisted Alexander polynomials, symplectic 4-manifolds and surfaces of minimal complexity

Stefan Friedl, Stefano Vidussi (2009)

Banach Center Publications

Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in [FV08a,FV08,FV07]. The results on surfaces of minimal complexity are new.

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