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We prove the following results on flag triangulations of 2-and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of ℝP and S × S have 11 and 12 vertices, respectively. In general, we show that 8 + 3k (resp. 8+4k) vertices suffice to obtain a flag triangulation of the connected sum of k copies of ℝP (resp. S × S ). In dimension 3, we describe an algorithm based on the Lutz–Nevo theorem which provides supporting computational evidence for the following generalization of the Charney–Davis conjecture: for any flag 3-manifold, γ := f −5 f +16 ≥ 16β , where f is the number of i-dimensional faces and β is the first Betti number over a field k. The conjecture is tight in the sense that for any value of β , there exists a flag 3-manifold for which the equality holds. 2 1 1 2 1 1 2 1 0 1 i 1 1

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