ContractibilityÌý of the space of tight contact structures on ${\mathbb R}^3$
ContractibilityÌý of the space of tight contact structures on ${\mathbb R}^3$
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Yasha Eliashberg, Stanford University/IAS
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30 years agoÌýI proved thatÌýany tight contact structure on theÌý3-sphere is diffeomorphic to the standard one. I also optimistically claimed at the sameÌý paperÌý that similar methods could be used to prove a multi-parametric version: the space of tight contact structures onÌý the 3-sphere, fixed at a point, is contractible.ÌýIn our recent joint with N. Mishachev paper weÌýproved this result.ÌýWhile the proof indeed roughly follows the strategyÌýofÌýmy 1991 paper, it is much more involved.ÌýIn particular, it uses a new criterion for tightnessÌýof of a characteristic foliation on the 2-sphere, which is valid without any contactÌý convexity assumptions.