# Contact structures and adjunction inequality in 3-manifolds

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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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
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