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Problems in Low Dimensional Contact Topology Geometric Topology Georgia International Topology Conference AMSIP Studies in Advanced Mathematics Volume Problems in Low Dimensional Contact Topology John B Etnyre and Lenhard L Ng During the Georgia International Topology Conference two problem ses sions were held concerning contact geometry The rst was run by Emmanuel Giroux and fo cused on problems in three dimensional contact top ology the second was run by John Etnyre and fo cused on Legendrian knots and contact homology This article collects problems from the sessions and adds some background Sec tions through deal with three dimensional contact geometry and Sections through deal with Legendrian knots We would like to thank the organizers of the Georgia Conference for running We would also the problem sessions and encouraging us to write this problem list like to thank Ko Honda and Will Kazez who to ok exceptional notes during the problem sessions and Josh Sablo who compiled a list of some of the problems discussed here I Three Dimensional Contact Geometry In this section the reader is assumed to b e familiar with the basic notions in contact geometry see and manifold top ology see Existence and types of contact structures Question Which manifolds have tight contact structures and what types of tight contact structures do they admit This question breaks down into several sub questions but rst note that a tight contact structure may b e weakly llable or weakly semillable strongly llable or strongly semillable holomorphically llable or holomorphically semillable universally tight or virtually overtwisted Recall that a holomorphically llable contact structure is also a strongly llable Moreover a weakly contact structure and strongly llable implies weakly llable Primary R secondary D M Mathematics Subject Classication Key words and phrases Contact structure tight Legendrian knot convex surface JE supp orted in part by NSF Grant DMS LN supp orted by the American Institute of Mathematics and NSF Grant DMS c American Mathematical So ciety and International Press JOHN B ETNYRE AND LENHARD L NG llable contact structure is tight It is known that there are tight contact structures that are not weakly llable and weakly llable contact structures that are not strongly llable the rst such examples were found by Eliashberg on the torus see also However on homology spheres we know that weakly llable structures must also b e strongly llable For a more thorough discussion of the notions of llability and their relations see Tight j Weakly symplecticall y semillable Strongly symplecticall y semillabl e Weakly symplectically llable Strongly symplecticall y llable Holomorphicall y llable If a contact structure is llable it is interesting to consider the number and type of llings In particular X Question Do there exist only nitely many strong or weak l lings of a xed M up to blowup and blowdown Actually it is not to o hard to show there are always innitely many strong llings if we allow the fundamental group of X to vary So it is most interesting to consider this question when X is required to b e simply connected The main technique used to show a contact structure is not llable is Seib erg Witten theory This technique was initiated by Lisca and only applies to manifolds with p ositive curvature It would b e exciting to nd other metho ds for showing that a contact structure is not llable Question How can one determine if a contact structure is l lable It is obvious that universally tight and virtually overtwisted contact structures form disjoint sets but it is not clear if all tight contact structures must fall into one of these sets For any manifold with a residually nite fundamental group it is known and not hard to prove that any tight contact structure is either universally tight or virtually overtwisted Recall that a group G is residually nite if every nontrivial element of G is in the complement of some nite index normal subgroup of G Of course it is conjectured that all manifolds have residually nite fundamental groups There are no known connections b etween a tight structure b eing llable in any sense and b eing universally tight or virtually overtwisted Question Are universal ly tight contact structures l lable An easier question might b e the following It is known that the converse is not true M is there some Question If is a universal ly tight contact structure on nite cover of M in which the pul ledback contact structure is l lable We recall that the Poincarehomology sphere with the nonstandard orientation P do es not admit a tight contact structure Moreover any tight contact structure on the connected sum M N can b e decomp osed into a tight contact PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY structure on M and a tight contact structure on N Thus any manifold of the form M P do es not admit any tight contact structure so when discussing existence of tight contact structures we will always consider irreducible manifolds If one b elieves the conjectured picture of manifolds coming from Thurstons Geometrization conjecture then the main question breaks into two Question Do al l hyperbolic manifolds admit a tight contact structure What type of tight structures do they support Question Which Seifert bered spaces admit tight contact structure Which are l lable Which are universal ly tight or virtual ly overtwisted Eliashberg and Thurston have shown that a Reebless foliation may b e p erturb ed into a tight in fact weakly semillable and universally tight contact structure Moreover Gabai can construct such a foliation on any irreducible manifold with b Thus when considering the ab ove questions one can fo cus on rational homology spheres or consider strong llability or virtually overtwisted contact structures Gompf has constructed strongly llable contact structures on many Seifert 2 3 1 b ered spaces Given the ab ove results here is our main concern Let 1 2 3 2 1 b e a Seifert b ered space over S with three singular b ers with invariants 1 2 3 2 3 1 if e then there is a strongly llable contact structure on 1 2 3 3 P i where e b c and bxc is the greatest integer less than or equal to x i i Moreover Gompf can also construct llable contact structures on many Seifert b ered spaces with e see The only manifold currently known not to admit a tight contact structure is P which is a Seifert manifold over S with invariants There are several other manifolds that do not admit weakly llable contact structures In Lisca has shown that the Seifert manifolds with invariants do not admit weakly llable contact structures It would b e very interesting to know Question Do the Seifert manifolds with invariants admit tight contact structures at al l It is b elieved at least by some that these manifolds do admit tight contact structures This result would b e particularly interesting b ecause then these would but not admitting b e the rst known manifolds admitting tight contact structures any llable structures existence of tight As for hyperb olic manifolds very little is known ab out the contact structures There are no known constructions of contact structures in terms of hyperb olic structures It was once conjectured that all hyperb olic manifolds admit taut foliations If this were true we could p erturb them into tight contact structures but it has recently b een shown that there are hyperb olic manifolds without taut foliations It is still p ossible however that all hyperb olic manifolds admit a tight contact structure Another imp ortant question is Question Do al l Haken manifolds admit a tight contact structure JOHN B ETNYRE AND LENHARD L NG Any manifold with b is known to b e Haken and from the work of Eliashberg Thurston and Gabai it is known that these manifolds admit semillable universally tight contact structures see also for a pro of that avoids foliation theory Thus our main interest here again is rational homology spheres Uniqueness and classication Of course the main problem here is Problem Classify tight contact structures on al l manifolds Currently contact structures are classied on the following manifolds S S S Lens spaces Torus bundles Circle bundles The classication is also known on some Seifert b ered spaces For example Tight or dep ending on orientation where Tight M denotes the set of tight contact structures up to isotopy See for further results on Seifert b ered spaces A sp ecic problem whose resolution would have great imp ortance for the clas sication of contact structures is Problem Classify tight contact structures on where is a surface of genus g In Honda Kazez and Maticclassify tight contact structures on e g when the relative Euler class of the contact structure is extremal and some mild conditions hold on the dividing curves It seems much more dicult to understand the situation when the Euler class is not extremal This problem has relevance to the classication of tight contact structures on surface bundles over circles manifolds with Haken decomp ositions and general manifolds given a tight contact The application of Problem to surface bundles is obvious and structure on a surface bundle split it along a convex b er the result is the dividing curves or characteristic foliation on the b oundary comp onents are related by the mono dromy of the bundle Now if one could normalize the dividing curves on a surface b er then one could apply
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