  Refining a basic result of Alexander, we show that two flag simplicial
complexes are piecewise linearly homeomorphic if and only if they can be
connected by a sequence of flag complexes, each obtained from the previous one
by either an edge subdivision or its inverse. For flag spheres we pose new
conjectures on their combinatorial structure forced by their face numbers,
analogous to the extremal examples in the upper and lower bound theorems for
simplicial spheres. Furthermore, we show that our algorithm to test the
conjectures searches through the entire space of flag PL spheres of any given
dimension.
