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Fixed link following Lukas Salchow's suggestion.
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Douglas Zare
Douglas Zare
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Overtwisted disks lift to overtwisted disks, so if $(M^3, \eta)$ is overtwisted then so is any cover.

The reverse is not true. There are tight contact structures with finite covers whichthat are overtwisted. If the universal cover is tight, the contact structure is called universally tight. There may be simpler references, but for examples of universally tight and virtually overtwisted contact structures, see Ko Honda, On the Classification of Tight Contact Structures II.On the Classification of Tight Contact Structures II. J. Differential Geometry Vol 55, Number 1 (2000), 83-143.

Overtwisted disks lift to overtwisted disks, so if $(M^3, \eta)$ is overtwisted then so is any cover.

The reverse is not true. There are tight contact structures with finite covers which are overtwisted. If the universal cover is tight, the contact structure is called universally tight. There may be simpler references, but for examples of universally tight and virtually overtwisted contact structures, see Ko Honda, On the Classification of Tight Contact Structures II. J. Differential Geometry Vol 55, Number 1 (2000), 83-143.

Overtwisted disks lift to overtwisted disks, so if $(M^3, \eta)$ is overtwisted then so is any cover.

The reverse is not true. There are tight contact structures with finite covers that are overtwisted. If the universal cover is tight, the contact structure is called universally tight. There may be simpler references, but for examples of universally tight and virtually overtwisted contact structures, see Ko Honda, On the Classification of Tight Contact Structures II. J. Differential Geometry Vol 55, Number 1 (2000), 83-143.

Source Link
Douglas Zare
Douglas Zare
  • 28k
  • 6
  • 90
  • 130

Overtwisted disks lift to overtwisted disks, so if $(M^3, \eta)$ is overtwisted then so is any cover.

The reverse is not true. There are tight contact structures with finite covers which are overtwisted. If the universal cover is tight, the contact structure is called universally tight. There may be simpler references, but for examples of universally tight and virtually overtwisted contact structures, see Ko Honda, On the Classification of Tight Contact Structures II. J. Differential Geometry Vol 55, Number 1 (2000), 83-143.