Flag complexes and homology

We prove several relations on the f-vectors and Betti numbers of flag complexes. For every flag complex Δ, we show that there exists a balanced complex with the same f-vector as Δ, and whose top-dimensional Betti number is at least that of Δ, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of Δ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the f-vector of Δ, in terms of the top-dimensional Betti number of Δ. This result has a continuous analog: If Δ is a (d−1)-dimensional flag complex whose (d−1)-th reduced homology group has dimension a≥0 (over some field), then the f-polynomial of Δ satisfies the coefficient-wise inequality f_Δ(x)≥(1+(ad+1)x)^d.