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All: I would appreciate any ideas, refs., etc. on the following:

Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the contact structure, say $\eta$ on $M^3$ lifts to a contact structure $\eta'$ on $X$.

Just hoping to get some information about the relationship between the tightness/overtwistedness and other contact properties of $(X, \eta')$ and the contact properties of $(M^3, \eta)$ . Can we, e.g., directly conclude from the tightness/overtwistedness of $(x, \eta')$ the tightness/overtwistedness of $(M^3 \eta)$?; are there any other known relationships between the contact properties of the two spaces? Maybe some properties of Legendrian knots and their twisting?

Thanks.

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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 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.

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    $\begingroup$ Incidentally, if $M$ is compact, residual finiteness implies that the universal cover is tight if and only if every finite cover is tight. $\endgroup$
    – Ian Agol
    Commented Jan 24, 2014 at 7:02
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    $\begingroup$ In fact, for $p>4$ most tight contact structures on $L(p,1)$ are Stein fillable but not universally tight. $\endgroup$
    – Marco Golla
    Commented Jan 24, 2014 at 8:37
  • $\begingroup$ The link is dead. Try this one math.ucla.edu/~honda/tight2.pdf. $\endgroup$
    – Lukas S
    Commented Oct 1, 2017 at 21:08

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