Algebraic Shifting Increases Relative Homology

. We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension. More precisely, let \Delta(K) denote the algebraically shifted complex of simplicial complex K, and let fi j (K; L) = dim k e H j (K; L; k) be the dimension of the jth reduced relative homology group over a field k of a pair of simplicial complexes L ` K. Then fi j (K; L) fi j (\Delta(K); \Delta(L)) for all j. The theorem is motivated by somewhat similar results about Grobner bases and generic initial ideals. Parts of the proof use Grobner basis techniques. 1. Introduction Algebraic shifting is a remarkable procedure that finds, for any simplicial complex K, a shifted (and hence combinatorially simpler) simplicial complex \Delta(K) with many of the same properties as K. For instance, the f-vector and homology Betti numbers are preserved; Bjorner and Kalai [BK1] used this fact to characterize the f-vectors and Betti numbers of simplicia..