#### Volume 14, issue 2 (2010)

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

### Klaus Niederkrüger and Francisco Presas

Geometry & Topology 14 (2010) 719–754
##### Abstract

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 $N×\left\{0\right\}$ in the model space $N×{ℝ}^{2k}$.

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.

##### Keywords
neighborhoods of contact submanifolds, fillability
Primary: 57R17
Secondary: 53D35
##### Publication
Received: 19 March 2009
Revised: 11 November 2009
Accepted: 5 November 2009
Published: 16 February 2010
Proposed: Leonid Polterovich
Seconded: Danny Calegari and Yasha Eliashberg
##### Authors
 Klaus Niederkrüger Klaus Niederkrüger Institut de mathématiques de Toulouse Université Paul Sabatier – Toulouse III 31062 Toulouse Cedex 9 France Francisco Presas Francisco Presas ICMAT CSIC Facultad de Matematicas Universidad Complutense de Madrid Plaza de Ciencias no 3 28040 Madrid Spain