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.
Bibliographical noteFunding Information:
Research of the first author was supported by the DFG Research Group "Polyhedral Surfaces", by Villum Fonden through the Experimental Mathematics Network and by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. Research of the second author was partially supported by Marie Curie grant IRG-270923 and ISF grant 805/11.