Chern slopes of simply connected complex surfaces of general type are dense in [2,3]

We prove that for any number $r$ in $[2,3]$, there are spin (resp. non-spin minimal) simply connected complex surfaces of general type $X$ with $c_1^2(X)\mathrm{/}{c}_2(X)$ arbitrarily close to $r$. In particular, this shows the existence of simply connected surfaces of general type arbitrarily close to the Bogomolov-Miyaoka-Yau line. In addition, we prove that for any $r\in [1,3]$ and any integer $q\ge 0$, there are minimal complex surfaces of general type $X$ with $c_1^2(X)\mathrm{/}{c}_2(X)$ arbitrarily close to $r$, and ${\pi }_1(X)$ isomorphic to the fundamental group of a compact Riemann surface of genus $q$.