Some remarks on the size of tubular neighborhoods in contact topology and fillability

Niederkrüger, Klaus¹ ; Presas, Francisco²

¹ Klaus Niederkrüger, Institut de mathématiques de Toulouse, Université Paul Sabatier – Toulouse III, 31062 Toulouse Cedex 9, France
² Francisco Presas, ICMAT, CSIC, Facultad de Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias no 3, 28040 Madrid, Spain

Geometry & topology, Tome 14 (2010) no. 2, pp. 719-754.

Résumé

The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold $N$ is uniquely determined by the contact structure on $N$, and the conformal symplectic structure of the normal bundle. In particular, if the submanifold $N$ has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of $N\times \left\{0\right\}$ in the model space $N\times {ℝ}^{2k}$.

In this article we make the observation that if $\left(N,{\xi }_N\right)$ is a $3$–dimensional overtwisted submanifold with trivial normal bundle in $\left(M,\xi \right)$, and if its model neighborhood is sufficiently large, then $\left(M,\xi \right)$ does not admit a symplectically aspherical filling.

Détail

DOI : 10.2140/gt.2010.14.719

Keywords: neighborhoods of contact submanifolds, fillability

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