Bigraded Betti numbers of some simple polytopes

Abstract

The bigraded Betti numbers b^{-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 b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology of the corresponding moment-angle manifold \mathcal Z_P, and therefore they find numerous applications in combinatorial commutative algebra and toric topology. Here we calculate some bigraded Betti numbers of the type \beta^{-i,2(i+1)} for associahedra, and relate the calculation of the bigraded Betti numbers for truncation polytopes to the topology of their moment-angle manifolds. These two series of simple polytopes provide conjectural extrema for the values of b^{-i,2j}(P) among all simple polytopes P with the fixed dimension and number of vertices.