Bigraded Betti numbers of certain simple polytopes

Abstract

The bigraded Betti numbers β−i,2j (P ) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P ]. The numbers β−i,2j (P ) reflect the combinatorial structure of P , as well as the topological structure of the corresponding moment-angle manifold ZP ; thus, they find numerous applications in combinatorial commutative algebra and toric topology. We calculate certain bigraded Betti numbers of the type β−i,2(i+1) for associahedra and apply the calculation of bigraded Betti numbers for truncation polytopes to study the topology of their moment-angle manifolds. Presumably, for these two series of simple polytopes, the numbers β−i,2j (P ) attain their minimum and maximum values among all simple polytopes P of fixed dimension with a given number of facets.