Hansjörg Geiges

A Legendrian surgery presentation of contact 3-manifolds

We prove that every closed, connected contact 3-manifold can be obtained from S 3 with its standard contact structure by contact (±1)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).

Symplectic fillability of tight contact structures on torus bundles

We study weak versus strong symplectic llability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically llable. For the 3{torus this theorem was established by Eliashberg. AMS Classication 53D35; 57M50, 57R65

Contact structures on 1-connected 5-manifolds

All manifolds in this paper are assumed to be closed, oriented and smooth. A contact structure on a (2 n + l)-dimensional manifold M is a maximally non-integrable hyperplane distribution D in the tangent bundle TM, i.e., D is locally denned as the kernel of a 1-form α satisfying α ۸ (da) n ۸ 0. A global form satisfying this condition is called a contact form. In the situations we are dealing with, every contact structure will be given by a contact form (see [5]). A manifold admitting a contact structure is called a contact manifold.

Handle moves in contact surgery diagrams

We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the classification of loose Legendrian knots. Along the way, we prove a one-to-one correspondence (up to Legendrian isotopy) between long Legendrian knots in 3-space and their completion in the 3-sphere.

Chapter 5 Contact geometry

Publisher Summary There are many excellent surveys covering specific aspects of contact geometry—for example, classification questions in dimension 3, dynamics of the Reeb vector field, various notions of symplectic fillability, transverse, and Legendrian knots and links. The chapter discusses some of the more fundamental differential topological aspects of contact geometry.

Symplectic cobordisms and the strong Weinstein conjecture

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

On the Topology of the Space of Contact Structures on Torus Bundles

We prove the existence of essential loops in the space of contact structures on torus bundles over the circle.

LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS

An apparatus for locating rectovaginal fistulas, including a cannula of a predetermined length, open-ended on its first end having an inflatable balloon portion on its closed second end; the first end of the cannula further comprising a cone portion along its length to seal up against the exterior wall entrance to the rectum, after the second end of the cannula has been inserted into a patients rectum; a fluid line is positioned within the cannula space for inflating the balloon after the cannula is in place in the rectum to seal the distal end of the rectum passage; a plurality of openings are provided along the wall of the cannula between the balloon and the cone at the first end of the cannula; a colored gel or other fluid insertable into the cannula to the extent that the gel or other fluid escapes the cannula through the openings and enters the void between the cannula and the wall of the rectum, with sufficient gel or other fluid entering the void so that gel or fluid would seep into any fistulas present and would be visually identifiable in the vagina for treatment.

Contact structures, equivariant spin bordism, and periodic fundamental groups

Abstract. Using contact surgery and equivariant bordism theory, we prove the existence of contact structures on all 5-dimensional spin manifolds with certain finite fundamental groups.

Contact Topology in Dimension Greater than Three

The aim of this talk is to give a survey of the known methods for constructing contact structures on manifolds of dimension greater than 3. We give an extensive list of contact manifolds that can be constructed via these methods, including some recent examples of contact structures on 5-dimensional manifolds found by a combination of contact surgery and cobordism theoretic techniques.