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On the equivalence of B-rigidity and C-rigidity for quasitoric manifolds. (English) Zbl 1316.57026

A quasitoric manifod \(M\) of dimension \(2n\) is a closed smooth \(2n\)-dimensional manifold with a locally standard action of an \(n\)-torus \(T^n\) such that the orbit space is a simple convex polytope \(P\). For a quasitoric manifold with orbit space \(P\), let \((\partial P)^\ast\) denote the dual of the boundary \(\partial P\) of \(P\). Two polytopes (or two simplicial complexes) are called combinatorially equivalent, if their face posets are isomorphic. Let \({\mathbf{k}}\) be a field of characteristic zero.
A simple convex polytope \(P\) is called cohomologically rigid if there exists a quasitoric manifold \(M\) over \(P\) and the following holds: Assume \(M'\) is another quasitoric manifold over a simple convex polytope \(P'\) such that as a ring \[ H^\ast(M;{\mathbf{k}})\cong H^\ast(M';{\mathbf{k}}). \] Then \(P\) and \(P'\) are combinatorially equivalent.
A simplicial complex \(K\) is called Buchstaber-rigid, if the following holds: Assume \(K'\) is another simplicial complex such that as a ring \[ H^\ast({\mathcal{Z}}_K; {\mathbf{k}})\cong H^\ast({\mathcal{Z}}_{K'};{\mathbf{k}}), \] where \({\mathcal{Z}}_K\) and \({\mathcal{Z}}_{K'}\) denote the moment-angle complexes of \(K\) and \(K'\), respectively. Then \(K\) and \(K'\) are combinatorially equivalent. Let then \(P\) be a simple convex polytope such that there is a quasitoric manifold over \(P\), and let \(K= (\partial P)^\ast\). The main result of this paper states that \(K\) is Buchstaber-rigid if and only if \(P\) is cohomologically rigid.

MSC:

57S15 Compact Lie groups of differentiable transformations
53C24 Rigidity results
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
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