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 \(2^{\Omega(k \log k)}\) and at most \(2^{O(k^{1.5} \log k)}\).