  We prove that among all flag 3-manifolds on $n$ vertices, the join of two
circles with $\left\lceil{\frac{n}{2}}\right \rceil$ and
$\left\lfloor{\frac{n}{2}}\right \rfloor$ vertices respectively is the unique
maximizer of the face numbers. This solves the first case of a conjecture due
to Lutz and Nevo. Further, we establish a sharp upper bound on the number of
edges of flag 5-manifolds and characterize the cases of equality. We also show
that the inequality part of the flag upper bound conjecture continues to hold
for all flag 3-dimensional Eulerian complexes and find all maximizers of the
face numbers in this class.
