Mazur manifolds and symplectic structures

We use the Heegaard Floer homology cobordism maps to obstruct the existence of a symplectic structure on the Akbulut-Kirby Mazur manifolds whose boundary is a Brieskorn sphere $Y$ among $\mathrm{\Sigma }(2,3,13),$ $\mathrm{\Sigma }(2,5,7)$ and $\mathrm{\Sigma }(3,4,5)$. Furthermore, we describe how our results imply the existence of exotic pairs of simply connected 4-manifolds, with definite intersection form, whose boundary is $Y$.