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.


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 k will denote a field of characteristic zero.A quasitoric manifold  of dimension 2 is a closed 2dimensional smooth manifold with a locally standard action of an -torus   := ( 1 )  whose orbit space is a simple convex polytope .The combinatorial structure of  can be decoded from the equivariant cohomology ring  *   (; k) of .The reason is that the equivariant cohomology ring of  is isomorphic to the Stanley-Reisner face ring k() of the dual  = () * of the boundary  of  and that the Stanley-Reisner face ring k() 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 k() contains certain significant information of a simple convex 2 International Journal of Mathematics and Mathematical Sciences polytope  supporting the quasitoric manifold, and the dimension of its Tor algebra gives rise to the bigraded Betti numbers  ,2 () which are purely combinatorial invariants of the polytope.It can be shown that the cohomology ring of the moment-angle complex Z  of the simplicial complex  = () * is isomorphic to Tor(k(); k) (refer to [1], Theorem 7.6, and see Section 2 for the definition of a moment-angle complex Z  ).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 Buchstaberrigid, 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 k() and its Tor-algebra Tor k[v] (k(), k) of k() and recall some well-known terminology used in this paper.In Section 3, we give a proof of our main Theorem 3.

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 wellknown 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 k where k[v] a denotes the vector space ⟨v a ⟩ over k spanned by v a .
The Stanley-Reisner ideal   of a simplicial complex  is defined as and its quotient ring k() = k[v]/  is called the Stanley-Reisner face ring of .
Since k() has finitely generated graded k[v]-module, there exists a free resolution of k() of length at most .Moreover, it has an N  -graded minimal free resolution as follows: where each homomorphism   is an N  -graded degreepreserving homomorphism.Now, applying the functor ⊗ k[v] k to the sequence (5), we can obtain the following chain complex of N  -graded k[v]-modules: where each homomorphism    is given by   ⊗ Id k .Since the free resolution ( 5) is minimal, the differentials    are actually zero homomorphisms.Hence the th homology module of the above chain complex, denoted Tor  k[v] (k(), k), is given by   ⊗ k[v] k.In particular, we have With these notations, by definition we have In case of a simple convex polytope , the bigraded Betti number  ,2 () is defined to be the bigraded Betti number  ,2 () 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 k() (refer to, e.g., [2], Section 5).The Krull dimension of k() is defined to be the maximal number of algebraically independent elements of k().Suppose that the Krull dimension of k() is .A sequence ( 1 ,  2 , . . .,   ) of homogeneous elements of k() is called a homogeneous system of parameters if the Krull dimension of k()/⟨ 1 ,  2 , . . .,   ⟩ is zero.A homogeneous system of parameters ( 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 {1, 2, . . ., }.Let  be an abstract simplicial complex on the vertex set [].For each simplex  ∈ , we set where and Then the moment-angle complex Z  on  is defined to be a subspace of ( 2 )  , as follows: When  = 2 [] , it is easy to see that Z  = ( 2 )  .On the other hand, when  = 2 ( 2 )  inherits a natural   -action whose orbit space is the unit cube   := [0, 1]  ⊂ R  ≥0 .This   -action on ( 2 )  then induces a canonical   -action on the moment-angle complex Z  .Refer to [1], Chapters 6 and 7 for more details on a moment-angle complex Z  .

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.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  * (Z  ; k) ≅ Tor(k(), k) 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  * (Z  ; k) and  * (Z   ; k).Thus  is combinatorially equivalent to   by the assumption that  is B-rigid.This implies that  is also combinatorially equivalent to   , which proves that  is Crigid.
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 k() and k(  ) 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 k() and k(  ) are isomorphic to each other as rings.To prove it, let  (resp.  ) denote the number of facets of  (resp.,   ).Since  1,2 () (resp.,  1,2 (  )) 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 N  -graded minimal free resolution for k() as follows: where each homomorphism   is an N  -graded degreepreserving homomorphism.Similarly, let be an N  -graded minimal resolution of k(  ).Since the free resolution of k() is minimal, for each  = 1, 2, . . ., ℎ we have In particular, this implies that ℎ is equal to ℎ  .Now, we show that there are N  -graded degreepreserving isomorphisms   (0 ≤  ≤ ℎ) from   to    so that the diagram (21) below commutes as follows: International Journal of Mathematics and Mathematical Sciences To do so, recall first that by ( 16 So it is now easy to see that   =    ∘   ∘  −1  is an N graded degree-preserving isomorphism from   to    which automatically makes the diagram (21) commute, as desired.
Moreover, by the standard argument using the diagramchasing we can also construct an N  -graded degreepreserving homomorphism  −1 between k() and k(  ) in such a way that the diagram (21) commutes.To be more precise, let  ∈ k().Then there is an element  ∈  0 such that  0 () = .We then define  −1 : k() → k(  ) by  −1 () =   0 ∘  0 ().Then it is well defined; that is, this definition is independent of the choice of .Indeed, let   ∈  0 such that  0 (  ) = .Then, since  0 ( −   ) = 0, there is an element  ∈  1 such that  −   =  1 ().Thus we have as desired.Finally, it is easy to see that by using the five-lemma ([10], p. 169) the ring homomorphism  −1 is also an N graded degree-preserving ring isomorphism between k() and k(  ).This, in particular, completes the proof of Lemma 5.
Recall now that the Stanley-Reisner face ring k() is given by k[v]/  , where   denotes the Stanley-Reisner ideal defined as ⟨v  |  ∉ ⟩.Let  be the ideal of k() generated by a regular sequence of homogeneous system of parameters of degree 2 elements.Then the cohomology ring  * (; k) 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 k().Then we have the following algebra isomorphism: The following lemma will also play an important role in the proof of Theorem 3.
By applying the same arguments as in the proof of Lemma 5 to N  -graded minimal free resolutions of k()/ and k(  )/  derived from the isomorphism (26) as in ( 16), it is now straightforward to see that two rings k()/ and k(  )/  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.
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.

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.
(k(  ), k) for each .On the other hand, since   is a ring isomorphism and  ,2 () is equal to  ,2 (  ) for all  and , this implies that actually   should be an N  -graded degree-preserving isomorphism from Tor k[v] (k(), k) to Tor  k[v] (k(  ), k) for each .As noted in Section 2, recall also that there are N  -graded degreepreserving isomorphisms   and    such that   :