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It is a theorem of Eliashberg that in a tight contact 3-manifold $(M, \xi)$ we have the adjunction inequality $|\langle e(s),[\Sigma] \rangle| \leq -\chi(\Sigma) $ where $s=s(\xi)$ is the spin$^c$-structure associated to $\xi$, $e(s)$ is its Euler class, and $\Sigma \subset M$ is an embedded surface, not the sphere.

The question is about a converse of this statement. Namely, if a spin$^c$ structure $s$ satisfies the above inequality for all surfaces $\neq S^2$, can we conclude that there is a tight contact structure representing $s$?

By the way, what can we say in general about such spin$^c$-structures?

P.s. For the sphere we must have $\langle e(s),[S^2] \rangle =0$.

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up vote 4 down vote accepted

Hi Daniele,

You can construct a counterexample like this. Call P the Poincaré homology sphere with reverse orientation. You know that Etnyre and Honda proved that there is no tight contact structure on P. Then take your favourite contact 3-manifold M, and consider the connected sum M#P. If there were a tight contact structure on M#P, it would restrict to a tight contact structure on P; this is easy to see: any contact structure is tight if and only if it is tight in the complement of a Darboux ball.

Since P is an integer homology sphere, there is a bijection between the Spin^c structures on M and the Spin^c structures on M#P. This means that, if M has a Spin^c structures satisfying Eliashberg's constraint, M#P has one too. However that Spin^c structure on M#P does not come from any tight contact structure.

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In spite of your answer, there's a sort of converse for $M$ irreducible. If $M$ is irreducible, this follows from Gabai's theorem that any Thurston norm minimizing surface may be extended to a taut orientable foliation. Then Eliashberg-Thurston implies the foliation may be perturbed to a tight contact structure realizing the euler class of the foliation, and satisfying the adjunction inequality. –  Ian Agol Jun 10 '13 at 0:43
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Yes, but this works for Spin^c structures which are extremal for the adjunction inequality. I don't know what happens in general if M is irreducible but the Spin^c structure is not extremal. In lens spaces one can find examples of Spin^c structures which do not come from tight contact structures, but this is a cheat because lens spaces are rational homology spheres and the adjunction inequality for them is empty. Unfortunately my box of examples contains only Seifert manifolds, which do not help much here. –  Paolo Ghiggini Jun 10 '13 at 4:15
    
That's interesting, thank you Paolo and Agol. It seems to me that the Agol's argument is correct for extremal spin$^c$-structures, but in general should fail (a simple counterexample: take a spin$^c$-structure with trivial Euler class, this exists in every 3-manifold, as I realized after posting the question, but this is maximally non-extremal). It could be nice to characterize spin$^c$-structures which come from tight contact structures. May be considering a Lefschetz fibration filling an open book compatible with $\xi$ for which it holds a 4-dimensional adjunction inequality? –  Daniele Zuddas Jun 10 '13 at 8:16
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