Cohomological rigidity of simple 3-polytopes with 10 facets

Abstract

A simple convex polytope $\it{P}$ is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $\it{P}$. A simple polytope $\it{Q}$ is irreducible if $\it{Q}$ cannot be expressed as a connected sum of more than 1 polytopes. It is known that irreducible 3-polytopes up to 9 facets are rigid. In this thesis, we investigate the cohomological rigidity of irreducible 3-polytopes with 10 facets by using computer programs $\it{plantri}$ and $\it{Macaulay2}$. The cohomological rigidity of $\it{P}$ is related to the $\it{bigraded Betti numbers}$ of $\it{its its Stanley-Reisner ring}$, important invariants coming from combinatorial commutative algebra.