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

Abstract

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 to be cohomologically rigid, or simply C-rigid, if the following two conditions hold.(i)There exists a quasitoric manifold over .(ii)Let be another quasitoric manifold over a simple convex polytope such that as a ring

Then is combinatorially equivalent to .

As mentioned above, the Stanley-Reisner face ring contains certain significant information of a simple convex polytope supporting the quasitoric manifold, and the dimension of its Tor algebra gives rise to the bigraded Betti numbers which are purely combinatorial invariants of the polytope. It can be shown that the cohomology ring of the moment-angle complex of the simplicial complex is isomorphic to (refer to [1], Theorem 7.6, and see Section 2 for the definition of a moment-angle complex ). Hence, it will be also natural to consider the following notion of rigidity, introduced first by Buchstaber in [3].

Definition 2. A simplicial complex is said to be Buchstaber-rigid, or simply B-rigid, if the following condition holds.(i)Let be another simplicial complex such that as a ring

Then is combinatorially equivalent to .

Note that there are simple convex polytopes supporting quasitoric manifolds which are not C-rigid such that their dual simplicial complexes are not B-rigid, either (refer to, e.g., [4], Example 1.1). Nonetheless, the aim of this paper is to show that B-rigidity is equivalent to C-rigidity in case of simple convex polytopes supporting quasitoric manifolds. This affirmatively answers a question in [4], Section 8, as follows.

Theorem 3. Let be a simple convex polytope supporting a quasitoric manifold, and let be the dual of the boundary of . Then B-rigidity of is equivalent to C-rigidity of .

We organize this paper as follows. In Section 2, we will collect some basic facts about the Stanley-Reisner face ring and its Tor-algebra of and recall some well-known terminology used in this paper. In Section 3, we give a proof of our main Theorem 3.

2. Stanley-Reisner Face Ring and Moment-Angle Complexes

The aim of this section is to set up some notations and briefly collect some basic material necessary for the proof of main Theorem 3 given in Section 3. In particular, we recall well-known facts about Stanley-Reisner face ring, following the works of [5, 6]. Refer to [1, 4] for more details and other notations used in this paper.

To do so, let be the polynomial algebra over on variables of degree 2. Let denote the set of nonnegative integers and . Each monomial in has the form . Thus is -graded, so that we have where denotes the vector space over spanned by .

The Stanley-Reisner ideal of a simplicial complex is defined as and its quotient ring is called the Stanley-Reisner face ring of .

Since has finitely generated graded -module, there exists a free resolution of of length at most . Moreover, it has an -graded minimal free resolution as follows: where each homomorphism is an -graded degree-preserving homomorphism.

Now, applying the functor to the sequence (5), we can obtain the following chain complex of -graded -modules: where each homomorphism is given by . Since the free resolution (5) is minimal, the differentials are actually zero homomorphisms. Hence the th homology module of the above chain complex, denoted , is given by . In particular, we have With these notations, by definition we have In case of a simple convex polytope , the bigraded Betti number is defined to be the bigraded Betti number of the simplicial complex that is dual to the boundary of .

Next, we recall the notion of a regular sequence of the Stanley-Reisner face ring (refer to, e.g., [2], Section 5). The Krull dimension of is defined to be the maximal number of algebraically independent elements of . Suppose that the Krull dimension of is . A sequence of homogeneous elements of is called a homogeneous system of parameters if the Krull dimension of is zero. A homogeneous system of parameters of is called regular if is not a zero divisor in . Equivalently, a sequence is regular if are algebraically independent and if is a finite dimensional free -module.

Finally, we close this section with reviewing the construction of a moment-angle complex associated with an abstract simplicial complex on a vertex set. To do so, let be a positive integer and let us denote by the set . Let be an abstract simplicial complex on the vertex set . For each simplex , we set where , , and Then the moment-angle complex on is defined to be a subspace of , as follows: When , it is easy to see that . On the other hand, when , where denotes the power set of , it can be easily shown that (refer to, e.g., [7], Example 2.4). Since as a subspace of is invariant under the standard action of on given by inherits a natural -action whose orbit space is the unit cube . This -action on then induces a canonical -action on the moment-angle complex . Refer to [1], Chapters 6 and 7 for more details on a moment-angle complex .

3. Proof of Theorem 3

The aim of this section is to give a proof of Theorem 3. Before doing it, we should remark that there seems to exist some confusing point at the end of [4], Section 8, where the authors erroneously claim the proof that C-rigidity implies B-rigidity for simple convex polytopes supporting quasitoric manifolds.

We begin with the following lemma.

Lemma 4. Let be a simple convex polytope supporting a quasitoric manifold , and let be the dual of the boundary of . If is B-rigid, then P is C-rigid.

Proof. To prove it, suppose that is B-rigid. Let be another simple convex polytope supporting a quasitoric manifold such that As before, let be the simplicial complex which is the dual of the boundary of the simple convex polytope . Then it follows from [4], Lemma 3.7, and Proposition 3.8 that as a ring Since we have a ring isomorphism for any simplicial complex by Buchstaber and Panov ([1], Theorem 7.6, and [8], Theorem 4.7), it follows from (14) that we have a ring isomorphism between and . Thus is combinatorially equivalent to by the assumption that is B-rigid. This implies that is also combinatorially equivalent to , which proves that is C-rigid.

Next, we show the following lemma.

Lemma 5. Let (resp. ) be a simple convex polytope supporting a quasitoric manifold (resp., ), and let (resp., ) be the dual of the boundary of (resp., ). Assume that as a ring Then two Stanley-Reisner face rings and are isomorphic to each other, as rings.

Proof. To prove it, note first that as a ring In particular, we have for all and .
Next, we claim that the Stanley-Reisner face rings and are isomorphic to each other as rings. To prove it, let (resp. ) denote the number of facets of (resp., ). Since (resp., ) is equal to the number of facets of (resp. ) by the nice formula for bigraded Betti numbers by Hochster ([9], Theorem 5.2 or [4], Theorem 3.3), it follows from (17) that should be equal to . Then, as in Section 2, consider an -graded minimal free resolution for as follows: where each homomorphism is an -graded degree-preserving homomorphism. Similarly, let be an -graded minimal resolution of . Since the free resolution of is minimal, for each we have In particular, this implies that is equal to .
Now, we show that there are -graded degree-preserving isomorphisms from to so that the diagram (21) below commutes as follows: To do so, recall first that by (16) there is a ring isomorphism from to for each . Hence, in particular, induces an isomorphism from to for each . On the other hand, since is a ring isomorphism and is equal to for all and , this implies that actually should be an -graded degree-preserving isomorphism from to for each . As noted in Section 2, recall also that there are -graded degree-preserving isomorphisms and such that So it is now easy to see that is an -graded degree-preserving isomorphism from to which automatically makes the diagram (21) commute, as desired.
Moreover, by the standard argument using the diagram-chasing we can also construct an -graded degree-preserving homomorphism between and in such a way that the diagram (21) commutes. To be more precise, let . Then there is an element such that . We then define by . Then it is well defined; that is, this definition is independent of the choice of . Indeed, let such that . Then, since , there is an element such that . Thus we have as desired.
Finally, it is easy to see that by using the five-lemma ([10], p. 169) the ring homomorphism is also an -graded degree-preserving ring isomorphism between and . This, in particular, completes the proof of Lemma 5.

Recall now that the Stanley-Reisner face ring is given by , where denotes the Stanley-Reisner ideal defined as . Let be the ideal of generated by a regular sequence of homogeneous system of parameters of degree elements. Then the cohomology ring is isomorphic to Then we need the following lemma ([1], Lemma 3.35, or [4], Lemma 3.6).

Lemma 6. Let be an ideal generated by a regular sequence of . Then we have the following algebra isomorphism:

The following lemma will also play an important role in the proof of Theorem 3.

Lemma 7. Under the same assumptions as in Lemma 5, two rings and are isomorphic to each other.

Proof. By Lemmas 5 and 6, we have an algebra isomorphism as follows: By applying the same arguments as in the proof of Lemma 5 to -graded minimal free resolutions of and derived from the isomorphism (26) as in (16), it is now straightforward to see that two rings and are isomorphic to each other, as desired.

Finally, we are ready to prove our main theorem of this section, as follows.

Theorem 8. Let be a simple convex polytope supporting a quasitoric manifold , and let be the dual of the boundary of . If is C-rigid, then is B-rigid.

Proof. To prove it, as before let (resp., ) be a simple convex polytope supporting a quasitoric manifold (resp. ), and let (resp., ) be the dual of the boundary of (resp., ). Assume further that as a ring By Lemma 7, we then have a ring isomorphism Since (resp., ) is a quasitoric manifold over a simple convex polytope (resp., ), it follows from the assumption of being C-rigid that is combinatorially equivalent to . So should be also combinatorial equivalent to , completing the proof of Theorem 8.

Proof of Theorem 3. The proof of Theorem 3 now follows immediately from Lemma 4 and Theorem 8.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the anonymous reader for valuable comments on this paper. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) (no. 2014001824).