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arXiv · 1101.0066

Bigraded Betti numbers of some simple polytopes

Ivan Limonchenko
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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.

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BibTeXRIS

Ivan Limonchenko. 2010-12-30. Bigraded Betti numbers of some simple polytopes. https://doi.org/10.1134/s000143461309006x

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