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 $\langle e(s),[S^2] \rangle =0$.



Contact structures and adjunction inequality in 3-manifolds
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. 


