On flag-no-square $4$-manifolds

Which $4$-manifolds admit a flag-no-square (fns) triangulation? We introduce the “star-connected-sum” operation on such triangulations, which preserves the fns property, from which we derive new constructions of fns $4$-manifolds. In particular, we show the following: (i) there exist non-aspherical fns $4$-manifolds, answering in the negative a question by Przytycki and Swiatkowski; (ii) for every large enough integer $k$ there exists a fns $4$-manifold $M_{2k}$ of Euler characteristic $2k$, and further, (iii) $M_{2k}$ admits a super-exponential number (in $k$) of fns triangulations - at least $2^{\mathrm{\Omega }(k\log k)}$ and at most $2^{O(k^{1.5}\log k)}$.