  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
$\mathbb{R} P^2$ and $\mathbb{S}^1\times \mathbb{S}^1$ 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
$\mathbb{R} P^2$ (resp. $\mathbb{S}^1\times \mathbb{S}^1$). 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, $\gamma_2:=f_1-5f_0+16\geq 16 \beta_1$,
where $f_i$ is the number of $i$-dimensional faces and $\beta_1$ is the first
Betti number over a field. The conjecture is tight in the sense that for any
value of $\beta_1$, there exists a flag 3-manifold for which the equality
holds.
