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 and have 11 and 12 vertices, respectively. In general, we show that (resp. ) vertices suffice to obtain a flag triangulation of the connected sum of copies of (resp. ). 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, , where is the number of -dimensional faces and is the first Betti number over a field . The conjecture is tight in the sense that for any value of , there exists a flag 3-manifold for which the equality holds.
"Minimal flag triangulations of lower-dimensional manifolds." Involve 13 (4) 683 - 703, 2020. https://doi.org/10.2140/involve.2020.13.683