Authors: Limonchenko, Ivan 
Title: Bigraded Betti numbers of certain simple polytopes
Journal: Mathematical Notes
Volume: 94
First page: 351
Last page: 363
Issue Date: 2013
Rank: M23
ISSN: 0001-4346
DOI: 10.1134/S000143461309006X
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.
Keywords: bigraded Betti numbers of a simple polytope | simple convex polytope | Stasheff polytope | associahedron | truncation polytope | stacked polytope | moment-angle manifold
Publisher: Springer Link

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