Repeated Lefschetz-like decompositions for flag doubly Cohen--Macaulay simplicial complexes and gamma vectors of flag spheres

Abstract: We find decompositions of $h$-polynomials of flag doubly Cohen-Macaulay simplicial complex that yield a direct connection between gamma vectors of flag spheres and constructions used to build them geometrically. More specifically, they are determined by iterated double suspensions and a "net nonnegative set of edge subdivisions" taking it to the given flag doubly Cohen-Macaulay simplicial complex. By a "net nonnegative set of edge subdivision", we mean a collection of edge subdivisions and contractions where there are at least as many edge subdivisions as contractions. Returning to the flag spheres, these repeated decompositions involve links over collections of disjoint edges and give an analogue of a Lefschetz map that applies to each step of the decomposition. The constructions used also give a direct interpretation of the Boolean decompositions coming from links and those of the entire simplicial complex. Roughly speaking, the Boolean vs. non-Boolean distinction is used to measure how far a flag sphere is from being the boundary of a cross polytope. An analogue of this statement for flag doubly Cohen-Macaulay simplicial complexes would replace boundaries of cross polytopes by repeated suspensions of links over edges of the given simplicial complex.