Constructions of open books and applications of convex surfaces in contact topology

Abstract

In the present thesis we introduce an extension of the contact connected sum, in the sense that we replace the tight $3$-balls by standard neighbourhoods of Legendrian graphs $G \subset (S^3,\xi_{st})$. By the use of convex surface theory we show that there is a Weinstein cobordism from the original contact manifold to the result of the extended contact connected sum. We approach the analogue of this result in higher dimensions, using different methods, and present a generalised symplectic $1$-handle which is used for the construction of exact symplectic cobordisms. Furthermore we describe compatible open books for the fibre connected sum along binding components of open books as well as for the fibre connected sum along multi-sections of open books. Given a Legendrian knot $L$ with standard neighbourhood $N$ in a closed contact $3$-manifold $(M,\xi)$, the homotopy type of the contact structure $\xi|_{M\setminus N}$ on the knot complement depends on the rotation number of $L$. We give an alternative proof of this folklore theorem, as well as for a second folklore theorem that states, up to stabilisation, the classification of Legendrian knots is purely topological. Let $\zeta$ denote the standard contact structure on the $3$-dimensional torus $T^3$. Denoting by $\Xi{(T^3,\zeta)}$ the connected component of $\zeta$ in the space of contact structures on $T^3$, we show that the fundamental group $\pi_1\big( \Xi{(T^3,\zeta)} \big)$ is isomorphic to $\mathbb{Z}$