Universität zu Köln

Dörner, Max Robert (2014). The Space of Contact Forms Adapted to an Open Book. PhD thesis, Universität zu Köln.

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Abstract

We construct a complete set of two consecutive obstructions against homotopies of pointed families of adapted contact forms with parameters in the n-sphere. Using these obstructions, we show that there is a manifold with an open book decomposition together with both infinitely many adapted contact forms that all induce the same Liouville form on one page but such that the underlying contact manifolds are not contactomorphic, and infinitely many non-homotopic adapted contact forms that all induce the same Liouville form on one page and such that the underlying contact manifolds are contactomorphic. Following this, we use the neighbourhood theorem for the binding of an open book decomposition that we introduce in the construction of the obstructions to construct special generalised caps of contact manifolds. This leads us to a proof that, on closed manifolds, the Reeb vector field of every contact form defining a contact structure supported by a tower of open book decompositions has a contractible orbit provided the binding in the lowest level of this tower embeds into a subcritical Stein manifold as a hypersurface of restricted contact type or is supported by an open book decomposition with trivial monodromy. Moreover, we show that the strong Weinstein conjecture holds for contact manifolds supported by an open book whose binding is planar.