On the Equivalence of B-Rigidity and C-Rigidity for Quasitoric Manifolds

For quasitoric manifolds and moment-angle complexes which are central objects recently much studied in toric topology, there are several important notions of rigidity formulated in terms of cohomology rings. The aim of this paper is to show that, among other things, Buchstaber-rigidity (or B-rigidity) is equivalent to cohomological-rigidity (or C-rigidity) for simple convex polytopes supporting quasitoric manifolds. 1. Introduction and Main Results In general, a cohomology ring of a given manifold is not enough to determine the manifold completely. However, there are some cases where we can characterize a given manifold in terms of a cohomology ring and which have recently attracted a great amount of attention in toric topology (see [1, 2]). For example, certain Bott manifolds and quasitoric manifolds, but not all of them, are such cases. The aim of this paper is, roughly speaking, to establish certain equivalence between two well-known notions of rigidity which essentially characterize quasitoric manifolds and also are formulated in terms of cohomology rings. In order to describe our results more precisely, we first need to collect some definitions and notations. To do so, throughout this paper will denote a field of characteristic zero. A quasitoric manifold of dimension is a closed -dimensional smooth manifold with a locally standard action of an -torus whose orbit space is a simple convex polytope . The combinatorial structure of can be decoded from the equivariant cohomology ring of . The reason is that the equivariant cohomology ring of is isomorphic to the Stanley-Reisner face ring of the dual of the boundary of and that the Stanley-Reisner face ring is in turn obtained by using certain combinatorial information of (refer to, e.g., [2], Theorem 4.8). In a similar vein, it is also expected that one can possibly obtain some information on a simple convex polytope from the usual cohomology ring of the manifold . If we have a quasitoric manifold over a simple convex polytope , from now on we will say that (or ) supports the quasitoric manifold , for simplicity. From these contexts, it is natural to give Definition 1. In order to explain it, recall first that the faces of a convex polytope form a face poset (or face lattice) where the partial ordering is by set containment of faces. Two polytopes are defined to be combinatorially isomorphic or combinatorially equivalent if their face posets are isomorphic (refer to [1], Section 1.1). An analogous definition obviously applies to two simplicial complexes. Definition 1. A simple convex polytope is said