1 Introduction
Problems surrounding cohomological rigidity have received a great deal of attention throughout the development of toric topology; for a recent overview, we recommend [Reference Fan, Ma and Wang16] and the references therein. Recall that a family
$\mathcal {C}$
of spaces (or of smooth manifolds) is called cohomologically rigid if, for all X,
$Y\in \mathcal {C}$
, a ring isomorphism
$H^\ast (X)\cong H^\ast (Y)$
implies that X and Y are homeomorphic (resp. diffeomorphic). Toric topology associates with each simplicial complex K with m vertices, the moment-angle complex
$\mathcal {Z}_K$
, a finite CW-complex with an action of the torus
$T^m=(S^1)^m$
, whose equivariant topology is intimately tied to the combinatorics of K and the homological algebra of its Stanley–Reisner ring. For example, if
$K=\partial P^*$
for some simple polytope P, then
has the structure of a smooth
$T^m$
-manifold. More generally, when the Stanley–Reisner ring
$\mathbb {Z}[K]$
is Gorenstein,
$\mathcal {Z}_K$
is a topological manifold called a moment-angle manifold. Computational evidence and a lack of counterexamples (despite the attention of many authors) makes a positive solution to the cohomological rigidity problem for this family of manifolds plausible.
An individual moment-angle manifold is said to be cohomologically rigid if its homeomorphism type among all moment-angle manifolds is distinguished by its cohomology ring. Since the combinatorial type of K determines
$\mathcal {Z}_K$
up to equivariant homeomorphism, one way to produce cohomologically rigid moment-angle manifolds is to find simplicial complexes K whose combinatorial type is uniquely determined by the ring
$H^\ast (\mathcal {Z}_K)$
.
Question 1.1 Let
$K_1$
and
$K_2$
be simplicial complexes, and let k be a field. When does a graded ring isomorphism
$H^\ast (\mathcal {Z}_{K_1};k)\cong H^\ast (\mathcal {Z}_{K_2};k)$
imply a combinatorial equivalence
$K_1\simeq K_2$
?
The question above is due to Buchstaber [Reference Buchstaber4], and simplicial complexes or simple polytopes for which the answer is positive are known as B-rigid (see Section 3 for precise definitions). Question 1.1 is reduced to a commutative algebraic problem by (1) the Stanley–Reisner correspondence between simplicial complexes and square-free monomial rings, and (2) an identification of
$H^\ast (\mathcal {Z}_K;k)$
with the Tor-algebra of the Stanley–Reisner ring
$k[K]$
(see Section 2).
B-rigid complexes seem to be rare in general (and indeed, combinatorially distinct polytopes often define diffeomorphic moment-angle manifolds, see [Reference Buchstaber, Erokhovets, Masuda, Panov and Park5, Example 3.4]), but examples to date have proven useful in establishing a variety of rigidity results for moment-angle manifolds, (quasi)toric manifolds, small covers, and other related spaces.
For example, flag simplicial
$2$
-spheres without chordless cycles of length
$4$
were shown to be B-rigid by Fan, Ma, and Wang [Reference Fan, Ma and Wang15], implying the cohomological rigidity of a large class of moment-angle manifolds including
$\mathcal {Z}_P$
for all fullerenes P (i.e., simple
$3$
-polytopes with only pentagonal and hexagonal facets). More generally, this class of simplicial
$2$
-spheres contains the dual simplicial complexes
$\partial P^*$
of all Pogorelov polytopes P, and their B-rigidity was used by Buchstaber et al. [Reference Buchstaber, Erokhovets, Masuda, Panov and Park5] to obtain cohomological rigidity results for all quasitoric manifolds and small covers over these
$3$
-polytopes.
Finite products of simplices are B-rigid by a result of Choi, Panov, and Suh [Reference Choi, Panov and Suh9]. This was used to show any quasitoric manifold M with
$H^\ast (M)\cong H^\ast (\prod _{i=1}^\ell \mathbb {C}P^{n_i})$
is in fact homeomorphic to
$\prod _{i=1}^\ell \mathbb {C}P^{n_i}$
[Reference Choi, Panov and Suh9, Theorem 1.5]. Quasitoric manifolds over products of simplices include an important family of toric varieties known as (generalized) Bott manifolds, and here B-rigidity implies that if a quasitoric manifold has cohomology isomorphic to that of a generalized Bott manifold, then their orbit spaces are combinatorially equivalent.
The B-rigidity of a prism (i.e., a product of a polygon and an interval) follows from [Reference Choi and Kim8], and this result was generalized to the product of a polygon and any simplex by Choi and Park in their study of projective bundles over toric surfaces [Reference Choi and Park11]. We will return to this example in Section 4.
These last examples motivate the question of whether B-rigid polytopes are closed under products. This question is also raised in work of Bosio [Reference Bosio2] where a weaker property for polytopes, puzzle-rigidity, is shown to be closed under products. The purpose of this short note is to answer this question affirmatively.
Theorem A The collection of B-rigid Gorenstein complexes is closed under finite joins. Consequently, the collection of B-rigid polytopes is closed under finite products.
This result substantially extends the class of known B-rigid polytopes, and subsumes those products established to be B-rigid in [Reference Choi and Kim8, Reference Choi, Panov and Suh9, Reference Choi and Park11]. We deduce Theorem A from the following more general structure theorem for moment-angle manifolds.
Theorem B Let K be a simplicial complex on m vertices, and let k be a field. If K is Gorenstein over k and the cohomology ring of
$\mathcal {Z}_K$
admits a tensor product decomposition
$$\begin{align*}H^\ast(\mathcal{Z}_K;k) \cong A_1\otimes\cdots\otimes A_\ell \end{align*}$$
as graded k-algebras, then there is a
$T^m$
-equivariant homeomorphism
$$\begin{align*}\mathcal{Z}_K \cong \mathcal{Z}_{K_1}\times\cdots\times \mathcal{Z}_{K_\ell} \end{align*}$$
with
$H^\ast (\mathcal {Z}_{K_i};k) \cong A_i$
for each
$i=1,\ldots ,\ell $
.
The remainder of the paper is organized as follows: In Section 2, we review the relevant background on simplicial complexes, Stanley–Reisner rings and moment-angle complexes and collect some facts concerning their cohomology rings. In Section 3, we define B-rigidity and prove Theorems A and B. In Section 4, we consider some consequences of the results above for quasitoric manifolds. The cohomology rings of these spaces are given by Artinian reductions of Stanley–Reisner rings, and the main results of this section (Theorems 4.3 and 4.7) address a variant of Question 1.1 that asks to what extent the orbit polytope of a quasitoric manifold M is determined by
$H^{\ast }(M)$
.
2 Preliminaries
Let k be a field. All graded algebras considered in this paper will be connected (that is, nonnegatively graded with
$A^0=k$
) and of finite type. The term commutative graded algebra will refer to a graded algebra that is commutative in the graded sense. All tensor products are taken over k.
2.1 Simplicial complexes and Stanley–Reisner rings
Let K be an abstract simplicial complex on the vertex set
$[m]=\{1,\ldots ,m\}$
. We will always assume that
$\varnothing \in K$
and that
$\{i\}\in K$
for all
$i\in [m]$
.
For a subset
$I\subseteq [m]$
, the full subcomplex of K on I is defined by
$$\begin{align*}K_I = \{\sigma\in K \,:\, \sigma\subseteq I\}. \end{align*}$$
A missing face of K is a minimal non-face, that is, a subset
$I\subseteq [m]$
with
$I\notin K$
and
$J\in K$
for every proper subset
$J\subset I$
(equivalently,
$K_I=\partial \Delta ^{|I|-1}$
). We will write
$\mathrm {MF}(K)$
for the set of missing faces of K.
The join of two simplicial complexes
$K_1$
and
$K_2$
on disjoint vertex sets is defined to be the simplicial complex
$$\begin{align*}K_1 \ast K_2 = \{\sigma\cup\tau \,:\, \sigma\in K_1,\, \tau\in K_2\}. \end{align*}$$
In particular,
$K\ast \Delta ^0$
is called the cone over K. Note that a simplicial complex on a given vertex set
$[m]$
is uniquely determined by its set of missing faces and that K decomposes as
$K=K_I\ast K_J$
if and only if the vertex set admits a partition
$[m]=I\sqcup J$
with the property that
$\mathrm {MF}(K)=\mathrm {MF}(K_I) \sqcup \mathrm {MF}(K_J)$
.
The polynomial algebra
$S=k[v_1,\ldots ,v_m]$
, having variables in bijection with the vertices of K, is naturally graded by
$\mathbb {Z}^m$
. We refer to this grading as the multidegree and use the notation
$\operatorname {mdeg}(v_1^{e_1}\!\ldots v_m^{e_m})=(e_1,\ldots ,e_m)$
.
The Stanley–Reisner ring of K is the multigraded algebra
$$\begin{align*}k[K]=S/\big(v_{i_1}\!\ldots v_{i_t} \,:\, \{i_1,\ldots,i_t\} \in \mathrm{MF }(K) \big). \end{align*}$$
The Koszul complex of the Stanley–Reisner ring is the differential graded algebra
$$ \begin{align} \operatorname{Kos}^{{k[K]}}(v_1,\ldots,v_m) = \big( k[K]\otimes \Lambda(u_1,\ldots,u_m),\ d\big), \quad d(u_i)=v_i, \end{align} $$
with its usual exterior algebra product and each
$u_i$
in homological degree
$1$
. The Tor-algebra
$$\begin{align*}\operatorname{Tor}^S_*(k[K],k) ={ H}_*\big(\operatorname{Kos}^{{k[K]}}(v_1,\ldots,v_m)\big) \end{align*}$$
inherits its product from the Koszul complex. It is naturally graded by
$\mathbb {Z}\times \mathbb {Z}^m$
; we write
$\operatorname {Tor}^S_i(k[K],k)_J$
for the subspace of homological degree i and multidegree J.
A multidegree
$J\in \mathbb {Z}^m$
is said to be square-free if each of its entries is
$0$
or
$1$
. The set of square-free multidegrees can be identified with the set of subsets of
$[m]$
, and we make this identification in the next result, which expresses
$\operatorname {Tor}^S_*(k[K],k)$
in terms of the reduced simplicial cohomology groups of full subcomplexes of K.
Theorem 2.1 (Hochster’s formula [Reference Hochster, McDonald and Morris17])
If J is square-free, then
$$\begin{align*}\operatorname{Tor}^S_i(k[K],k)_J\cong \widetilde{H}^{|J|-i-1}(K_J;k), \end{align*}$$
with the convention that
$\widetilde {H}^{-1}(K_\varnothing ;k)=k$
. Otherwise
$\operatorname {Tor}^S_i(k[K],k)_J=0$
.
Remark 2.2 Hochster’s formula allows us to identify multidegrees of
$\operatorname {Tor}^S_\ast (k[K],k)$
with subsets of
$[m]$
, and we will do this henceforth. It follows as well that for any subset
$J\subseteq [m]$
the graded subspace
$$\begin{align*}\operatorname{Tor}^S_\ast(k[K],k)_{\subseteq J}= \bigoplus_{I\subseteq J}\operatorname{Tor}^S_\ast(k[K],k)_I \end{align*}$$
is a subalgebra of
$\operatorname {Tor}^S_\ast (k[K],k)$
, and moreover that the natural projection
$$\begin{align*}\mathrm{proj}_{\subseteq J}\colon \operatorname{Tor}^S_\ast(k[K],k) \longrightarrow \operatorname{Tor}^S_\ast(k[K],k)_{\subseteq J} \end{align*}$$
is an algebra homomorphism.
Lemma 2.3 In homological degree
$1$
, a k-basis for
$\operatorname {Tor}_{1}^S(k[K],k)$
is given by
$$\begin{align*}\left\{ [v_{i_1}\!\ldots v_{i_t}u_{i_{t+1}}] \,:\, \{i_1,\ldots,i_{t+1}\} \in \mathrm{MF }(K) \right\}. \end{align*}$$
Proof A direct computation with the Koszul complex shows that
$v_{i_1}\!\ldots v_{i_t}u_{i_{t+1}}$
is closed and not exact when
$\{i_1,\ldots ,i_{t+1}\} \in \mathrm {MF}(K)$
. The resulting nonzero classes
$[v_{i_1}\!\ldots v_{i_t}u_{i_{t+1}}]$
are linearly independent since they lie in distinct multidegrees, and therefore form a basis since
$\dim _k\operatorname {Tor}_{1}^S(k[K],k)=|\mathrm {MF}(K)|$
, the minimal number of generators of the Stanley–Reisner ideal.
2.2 Moment-angle complexes and their cohomology rings
We now recall some notions from toric topology, largely in order to lay out our notation. Fix as before a simplicial complex K on a vertex set
$[m]$
. Identify
$D^2$
and
$S^1$
with the unit disk in
$\mathbb {C}$
and its boundary, respectively.
The moment-angle complex over K is the subspace of the polydisk
$(D^2)^m$
defined by
$$\begin{align*}\mathcal{Z}_K = \bigcup_{\sigma\in K} (D^2,S^1)^\sigma \subseteq (D^2)^m, \end{align*}$$
where
$$\begin{align*}(D^2,S^1)^\sigma=\left\{ (z_1,\ldots,z_m)\in (D^2)^m \,:\, z_i\in S^1\text{ if }i\notin \sigma\right\}. \end{align*}$$
The coordinatewise action of the m-torus
$T^m=(S^1)^m$
on
$(D^2)^m$
restricts to an action of
$T^m$
on
$\mathcal {Z}_K$
. The equivariant cohomology ring
$H^\ast _{T^m}(\mathcal {Z}_K;k)$
is naturally a graded module over
$H^\ast (BT^m;k)\cong S$
(with polynomial generators
$v_1,\ldots ,v_m$
in cohomological degree
$2$
), and there is an isomorphism of graded S-modules
$$\begin{align*}H^\ast_{T^m}(\mathcal{Z}_K;k) \cong k[K], \end{align*}$$
see, for example, [Reference Buchstaber and Panov6, Corollary 4.3.3].
For the ordinary cohomology, the cellular cochain complex of
$\mathcal {Z}_K$
can be shown to be quasi-isomorphic to the Koszul complex (1), which leads to the following fundamental result.
Theorem 2.4 [Reference Buchstaber and Panov6, Theorem 4.5.4]
There is an isomorphism of graded algebras
$$\begin{align*}H^\ast(\mathcal{Z}_K;k) \cong \operatorname{Tor}^S_\ast(k[K],k), \end{align*}$$
where
$$\begin{align*}H^n(\mathcal{Z}_K;k) \cong \bigoplus_{n=2|J|-i} \operatorname{Tor}^S_i(k[K],k)_J. \end{align*}$$
In particular, the cohomology ring of
$\mathcal {Z}_K$
obtains a multigrading from the Tor-algebra. Under the isomorphism of Theorem 2.4, the projection operators
$$\begin{align*}\mathrm{proj}_{\subseteq J}\colon \operatorname{Tor}^S_\ast(k[K],k) \longrightarrow \operatorname{Tor}^S_\ast(k[K],k)_{\subseteq J} \end{align*}$$
of Remark 2.2 can be identified with homomorphisms
$$\begin{align*}j^\ast\colon H^\ast(\mathcal{Z}_K;k) \longrightarrow H^\ast(\mathcal{Z}_{K_J};k) \end{align*}$$
induced by the natural maps
$j\colon \mathcal {Z}_{K_J} \to \mathcal {Z}_K$
coming from inclusions of full subcomplexes
$K_J \to K$
.
Definition 2.5 A commutative graded k-algebra A is a Poincaré duality algebra of dimension n if it is finite dimensional, and if the bilinear forms
$$\begin{align*}A^i\otimes A^{n-i} \to k, \quad a\otimes b \mapsto \epsilon(ab) \end{align*}$$
are nondegenerate for some homogeneous map
$\epsilon \colon A \to k$
of degree
$-n$
.
Lemma 2.6 [Reference Eisenbud13, Proposition 21.5]
A finite dimensional, connected, graded algebra A is Poincaré duality of dimension n if and only if the socle
$\mathrm {soc}(A)=\mathrm {ann}_A(A_{\geqslant 1})$
is one-dimensional and concentrated in degree n. In this case, the orientation map
$\epsilon \colon A\to k$
corresponds to a choice of isomorphism
$\mathrm {soc}(A)\to k$
.
Lemma 2.7 Let A be a Poincaré duality algebra of dimension n. If there is an isomorphism
$A \cong A_1\otimes A_2$
of graded algebras, then
$A_1$
and
$A_2$
are Poincaré duality algebras of dimension
$n_1$
and
$n_2$
, respectively, with
$n_1+n_2=n$
.
Proof Since
$A \cong A_1\otimes A_2$
implies
$\mathrm {soc}(A)\cong \mathrm {soc}(A_1)\otimes \mathrm {soc}(A_2)$
, the socle of A is one-dimensional if and only if so are the socles of
$A_1$
and
$A_2$
.
A classical result of Avramov and Golod [Reference Avramov and Golod1] characterizes Poincaré duality for the Koszul homology of a local ring in terms of the Gorenstein property. The simplicial complex K is called Gorenstein over k if the Stanley–Reisner ring
$k[K]$
is a Gorenstein ring, and K is simply called Gorenstein if it is Gorenstein over every field k. Moreover, K is called Gorenstein
${}^*$
if it is Gorenstein and K is not a cone (i.e.,
$K=\mathrm {core}(K)$
).
Combined with Theorem 2.4, a graded version of Avramov–Golod’s theorem (see [Reference Bruns and Herzog3, Theorem 3.4.5]) leads to the following characterization of Poincaré duality for moment-angle complexes.
Theorem 2.8 [Reference Buchstaber and Panov6, Theorem 4.6.8]
The cohomology ring
$H^\ast (\mathcal {Z}_K;k)$
is a Poincaré duality algebra if and only if K is Gorenstein over k.
Remark 2.9 The topology behind this characterization is clarified by a result of Cai which states that
$\mathcal {Z}_K$
is a closed topological
$(m+n)$
-manifold precisely when K is a generalized homology
$(n-1)$
-sphere [Reference Cai7, Corollary 2.10], which in turn is equivalent to the condition that K is Gorenstein
${}^*$
by a well-known result of Stanley [Reference Stanley19].
Gorenstein
${}^*$
complexes of particular importance in toric topology arise from simple polytopes as follows. Let P be a simple (convex) polytope of dimension n, and denote the facets (faces of dimension
$n-1$
) of P by
$F_1,\ldots ,F_m$
. The simplicial complex K dual to P has vertex set
$[m]=\{1,\ldots ,m\}$
and
$J\subseteq [m]$
belongs to K if and only if
$\bigcap _{i\in J}F_i\neq \varnothing $
. Equivalently, K is the boundary of the polytope dual to P, i.e.,
$K=\partial P^*$
.
If
$K'$
is the simplicial complex dual to another simple polytope
$P'$
, then it is easy to see that the simplicial complex dual to the product
$P\times P'$
is the join
$K\ast K'$
.
Through this construction, we associate with P the Stanley–Reisner ring and moment-angle manifold
$$\begin{align*}k[P]=k[K] \quad\textrm{ and }\quad \mathcal{Z}_P=\mathcal{Z}_K. \end{align*}$$
In particular, if
$k[P]$
is generated by
$v_1,\ldots ,v_m$
as in Section 2.1, then by Theorem 2.4
$$\begin{align*}H^*(\mathcal{Z}_P;k)\cong H^*\big(\operatorname{Kos}^{{k[P]}}(v_1,\ldots,v_m)\big). \end{align*}$$
Moment-angle manifolds corresponding to simple polytopes can be given smooth structures (for which the
$T^m$
-action is smooth) by identifying
$\mathcal {Z}_P$
with a nonsingular intersection of quadrics in
$\mathbb {C}^m$
(see [Reference Buchstaber and Panov6, Section 6.1] for details).
3 B-rigidity
In this section, we prove Theorem B (Theorem 3.2) and use it to derive some corollaries, including Theorem A (Corollary 3.5). We begin with a definition.
A simplicial complex K with
$K=\mathrm {core}(K)$
is called B-rigid over k if for every
$K'$
with
$K'=\mathrm {core}(K')$
, a graded ring isomorphism
$H^\ast (\mathcal {Z}_K;k)\cong H^\ast (\mathcal {Z}_{K'};k)$
implies a combinatorial equivalence
$K\simeq K'$
. We simply say that K is B-rigid if it is B-rigid over every field k.
Remark 3.1
B-rigidity was introduced and defined as above in [Reference Choi, Panov and Suh9] to address Question 1.1 (cf. [Reference Buchstaber4, Lecture IV and Problem 7.6]), and has since been studied in [Reference Buchstaber, Erokhovets, Masuda, Panov and Park5, Reference Choi and Park10, Reference Erokhovets, Rokhlin-Memorial, Buchstaber, Malyutin and Vershik14–Reference Fan, Ma and Wang16], where some variations on the definition above have appeared. For example, in [Reference Fan, Ma and Wang16], bigraded isomorphisms of cohomology rings are used to define B-rigidity while complexes satisfying the definition above are called strongly B-rigid. In [Reference Fan, Ma and Wang15], a complex is called B-rigid when it is B-rigid over
$\mathbb {Z}$
in the sense above. We note that any complex satisfying the definition above is also B-rigid in the bigraded and integral senses.
In [Reference Buchstaber, Erokhovets, Masuda, Panov and Park5, Reference Choi and Park10, Reference Erokhovets, Rokhlin-Memorial, Buchstaber, Malyutin and Vershik14], the notion of B-rigidity is defined for simple polytopes rather than simplicial complexes. A simple polytope P is called B-rigid over k if for every simple polytope
$P'$
, a graded ring isomorphism
$H^\ast (\mathcal {Z}_P;k)\cong H^\ast (\mathcal {Z}_{P'};k)$
implies a combinatorial equivalence
$P\simeq P'$
. As before, we say that P is B-rigid if it is B-rigid over every field k. Our main results apply in the more general context of simplicial complexes.
Theorem 3.2 Let K be a simplicial complex on vertex set
$[m]$
, and let k be a field. If K is Gorenstein over k and the cohomology ring of
$\mathcal {Z}_K$
admits a tensor product decomposition
$$\begin{align*}H^\ast(\mathcal{Z}_K;k) \cong A_1\otimes\cdots\otimes A_\ell \end{align*}$$
as graded k-algebras, then there is a
$T^m$
-equivariant homeomorphism
$$\begin{align*}\mathcal{Z}_K \cong \mathcal{Z}_{K_1}\times\cdots\times \mathcal{Z}_{K_\ell}, \end{align*}$$
where
$H^\ast (\mathcal {Z}_{K_i};k) \cong A_i$
for each
$i=1,\ldots ,\ell $
.
Proof Note that if
$K=K'\ast \Delta ^0$
, then
$\mathcal {Z}_K$
is
$T^m$
-equivariantly homeomorphic to
$\mathcal {Z}_{K'} \times \mathcal {Z}_{\Delta ^0}=\mathcal {Z}_{K'} \times (D^2)$
with
$T^m=T^{m-1}\times S^1$
acting coordinatewise on the product. We may therefore assume without loss of generality that K is not a cone and hence that K is Gorenstein
${}^*$
over k.
It will suffice to assume
$\ell =2$
, so suppose
$$\begin{align*}\varphi\colon A_1\otimes A_2 \longrightarrow H^\ast(\mathcal{Z}_K;k) \end{align*}$$
is an isomorphism of graded k-algebras. Since K is Gorenstein
${}^*$
over k, the cohomology ring
$H^\ast (\mathcal {Z}_K;k)$
is a Poincaré duality algebra by Theorem 2.8, and so are
$A_1$
and
$A_2$
by Lemma 2.7. Let
$\tau _i \in A_i$
be a generator of the highest degree nonzero graded component of
$A_i$
,
$i=1,2$
. Then
generates the top cohomology group of
$\mathcal {Z}_K$
, and it follows from the Gorenstein
${}^*$
property that
$\tau $
is homogeneous of multidegree
$\operatorname {mdeg}(\tau )=[m]$
. Identifying
$H^\ast (\mathcal {Z}_K;k)$
with
$\bigoplus _{J\subseteq [m]} \mathrm {Tor}_\ast ^S(k[K],k)_J$
using Theorem 2.4, write
$$\begin{align*}\varphi(\tau_i) = {\textstyle \sum_{J\subseteq [m]}} \varphi(\tau_i)_J, \quad \varphi(\tau_i)_J\in \mathrm{Tor}_\ast^S(k[K],k)_J \end{align*}$$
for
$i=1,2$
and choose multidegrees
$U,V\subseteq [m]$
such that
$\varphi (\tau _1)_U \cdot \varphi (\tau _2)_V = c\tau $
for some nonzero
$c\in k$
. Note that
$U\cap V=\varnothing $
by square-freeness (Theorem 2.1), and
$U\cup V=[m]$
since
$\operatorname {mdeg}(\tau )=[m]$
.
Next, define subalgebras of
$H^\ast (\mathcal {Z}_K;k)$
by
$$\begin{align*}\widetilde{A}_1 = \mathrm{proj}_{\subseteq U}(\varphi(A_1))\quad\text{and}\quad \widetilde{A}_2 = \mathrm{proj}_{\subseteq V}(\varphi(A_2)) \end{align*}$$
using the projections from Remark 2.2. Since
$\mathrm {proj}_{\subseteq U} \varphi (\tau _1) \neq 0$
, and every nonzero element of
$\varphi (A_1)$
divides
$\varphi (\tau _1)$
by Poincaré duality, it follows that
$\mathrm {proj}_{\subseteq U}|_{\varphi (A_1)}$
is injective and therefore defines an isomorphism
$\varphi (A_1) \cong \widetilde {A}_1$
. Similarly,
$\varphi (A_2) \cong \widetilde {A}_2$
. Consider the homomorphism of graded algebras
$\widetilde {A}_1 \otimes \widetilde {A}_2 \to H^\ast (\mathcal {Z}_K;k)$
defined by multiplication. Since
$\widetilde {A}_1 \otimes \widetilde {A}_2$
is clearly a Poincaré duality algebra with socle generated by
$\varphi (\tau _1)_U \cdot \varphi (\tau _2)_V$
, this map is injective and therefore an isomorphism as
$\dim _k(\widetilde {A}_1 \otimes \widetilde {A}_2) =\dim _k(A_1 \otimes A_2) =\dim _k H^\ast (\mathcal {Z}_K;k)$
.
Finally, in homological degree
$1$
, observe that the decomposition
$H^\ast (\mathcal {Z}_K;k) \cong \widetilde {A}_1 \otimes \widetilde {A}_2$
implies that each indecomposable generator
$$\begin{align*}[v_{i_1}\!\ldots v_{i_t}u_{i_{t+1}}] \in \mathrm{Tor}_{1}^S(k[K],k) \subseteq H^\ast(\mathcal{Z}_K;k) \end{align*}$$
of Lemma 2.3 must lie either in
$\widetilde {A}_1$
or
$\widetilde {A}_2$
(since it is homogeneous of multidegree
$\{i_1,\ldots ,i_{t+1}\}$
and
$\widetilde {A}_1 \subseteq \operatorname {Tor}^S_\ast (k[K],k)_{\subseteq U}$
,
$\widetilde {A}_2 \subseteq \operatorname {Tor}^S_\ast (k[K],k)_{\subseteq V}$
). In particular, each missing face
$\sigma =\{i_1,\ldots ,i_{t+1}\} \notin K$
satisfies either
$\sigma \in U$
or
$\sigma \in V$
, that is,
$\mathrm {MF}(K)=\mathrm {MF}(K_U) \sqcup \mathrm {MF}(K_V)$
. It follows that
$K=K_U\ast K_V$
and hence
$\mathcal {Z}_K$
is equivariantly homeomorphic to
$\mathcal {Z}_{K_U} \times \mathcal {Z}_{K_V}$
. Note that since
$H^\ast (\mathcal {Z}_{K_U};k)\cong \operatorname {Tor}^S_\ast (k[K],k)_{\subseteq U}$
and
$H^\ast (\mathcal {Z}_{K_V};k)\cong \operatorname {Tor}^S_\ast (k[K],k)_{\subseteq V}$
, both inclusions
$\widetilde {A}_1 \subseteq \operatorname {Tor}^S_\ast (k[K],k)_{\subseteq U}$
and
$\widetilde {A}_2 \subseteq \operatorname {Tor}^S_\ast (k[K],k)_{\subseteq V}$
must be equalities for dimension reasons. Therefore
$$\begin{align*}H^\ast(\mathcal{Z}_{K_U};k)\cong \widetilde{A}_1 \cong \varphi(A_1)\cong A_1, \end{align*}$$
and similarly
$H^\ast (\mathcal {Z}_{K_V};k)\cong A_2$
, as desired.
The next corollary follows immediately from the proof above. It is interesting to note that any product decomposition of a moment-angle manifold up to homotopy can be improved to one up to
$T^m$
-equivariant homeomorphism.
Corollary 3.3 Let K be a Gorenstein simplicial complex on the vertex set
$[m]$
. The following conditions are equivalent:
-
(a)
$H^\ast (\mathcal {Z}_K)$
decomposes nontrivially as a tensor product. -
(b)
$\mathcal {Z}_K$
is homotopy equivalent to a product of non-contractible spaces. -
(c)
$\mathcal {Z}_K$
is
$T^m$
-equivariantly homeomorphic to a product of non-contractible moment-angle manifolds. -
(d)
$K=K_1\ast K_2$
, where
$K_i$
is not a simplex,
$i=1,2$
.
Remark 3.4 Of course, for a moment-angle manifold over a simple polytope P, it follows from the results above that a nontrivial tensor product decomposition of
$H^\ast (\mathcal {Z}_P;k)$
implies a product decomposition of P. In this case, the
$T^m$
-equivariant homeomorphisms of Theorem 3.2 and Corollary 3.3 can be replaced by
$T^m$
-equivariant diffeomorphisms.
Corollary 3.5 The collection of B-rigid Gorenstein complexes is closed under finite joins. Consequently, the collection of B-rigid polytopes is closed under finite products.
Proof Let
$K_1$
and
$K_2$
be B-rigid Gorenstein complexes. To see that
$K_1\ast K_2$
is B-rigid, assume there is a graded ring isomorphism
$H^\ast (\mathcal {Z}_{K_1\ast K_2};k)\cong H^\ast (\mathcal {Z}_L;k)$
for some complex L with
$L=\mathrm {core}(L)$
. Then, since
$H^\ast (\mathcal {Z}_{K_1\ast K_2};k)\cong H^\ast (\mathcal {Z}_{K_1};k)\otimes H^\ast (\mathcal {Z}_{K_2};k)$
, the proof of Theorem 3.2 implies that L decomposes as a join
$L_1\ast L_2$
with
$H^\ast (\mathcal {Z}_{L_i};k)\cong H^\ast (\mathcal {Z}_{K_i};k)$
for
$i=1,2$
. It now follows from the B-rigidity of
$K_i$
that there is a combinatorial equivalence
$K_i\simeq L_i$
for
$i=1,2$
, and hence
$L\simeq K_1\ast K_2$
. Therefore
$K_1\ast K_2$
is B-rigid.
4 Quasitoric manifolds
In their foundational work [Reference Davis and Januszkiewicz12], Davis and Januszkiewicz introduced the notion of a quasitoric manifold as a topological generalization of a nonsingular projective toric variety (or toric manifold). In this section, we discuss some implications of the results of the previous section for quasitoric manifolds.
Let P be a simple convex polytope of dimension n. A quasitoric manifold over P is a closed smooth
$2n$
-dimensional manifold M equipped with a smooth locally standard action of
$T^n$
such that the orbit space
$M/T^n$
can be identified with P. (A locally standard
$T^n$
-action is one which is locally modeled on the standard action of
$T^n$
on
$\mathbb {C}^n$
, and this property implies that the orbit space is a manifold with corners.)
Every quasitoric manifold over P arises as a quotient
$M\cong \mathcal {Z}_P/T^{m-n}$
for some subtorus
$T^{m-n}\subseteq T^m$
that acts freely on
$\mathcal {Z}_P$
[Reference Buchstaber and Panov6, Proposition 7.3.12], resulting in a principal fibration sequence
$$\begin{align*}T^{m-n} \longrightarrow \mathcal{Z}_P \longrightarrow M. \end{align*}$$
The cohomology of quasitoric manifolds was described in [Reference Davis and Januszkiewicz12]. Associated with the
$T^n$
-action on M, there is a regular sequence
$t_1,\ldots ,t_{n}$
of linear elements in
$k[P]$
such that
$$ \begin{align} H^\ast(M;k)\cong k[P]/(t_1,\ldots,t_{n}). \end{align} $$
Since
$t_1,\ldots ,t_n$
is a regular sequence, it follows from Theorem 2.4 that there is an isomorphism (cf. [Reference Buchstaber and Panov6, Lemma A.3.5])
$$ \begin{align} H^\ast(\mathcal{Z}_P;k)\cong \operatorname{Tor}^{S/\mathcal{J}}_\ast(k[P]/\mathcal{J},k), \end{align} $$
where
$\mathcal {J}$
denotes the ideal generated by (homogeneous lifts to S of)
$t_1,\ldots ,t_n$
.
As a consequence of the isomorphisms (2) and (3), we emphasize that the cohomology ring of any quasitoric manifold over P determines, in particular, the cohomology ring of the moment-angle manifold over P.
Lemma 4.1 [Reference Choi, Panov and Suh9, Lemma 3.7]
Let M and
$M'$
be quasitoric manifolds over simple polytopes P and
$P'$
, respectively. If there is a graded ring isomorphism
$H^\ast (M;k)\cong H^\ast (M';k)$
, then there is a graded ring isomorphism
$H^\ast (\mathcal {Z}_P;k)\cong H^\ast (\mathcal {Z}_{P'};k)$
.
Remark 4.2 The cohomology ring of a quasitoric manifold M contains more information about its orbit polytope P than the ring
$H^\ast (\mathcal {Z}_P)$
does, but it does not determine the combinatorial type of P in general. A simple polytope P is called C-rigid (over k) if its combinatorial type is determined by
$H^\ast (M;k)$
for any quasitoric manifold M over P—that is, if for every quasitoric manifold
$M'$
over
$P'$
, a graded ring isomorphism
$H^\ast (M;k)\cong H^\ast (M';k)$
implies a combinatorial equivalence
$P\simeq P'$
(cf. [Reference Buchstaber4, Definition A.11], [Reference Choi, Panov and Suh9, Definition 1.2]). By Lemma 4.1, all B-rigid polytopes are C-rigid. Examples of C-rigid polytopes that are not B-rigid are given in [Reference Choi and Park10], while the
$3$
-polytopes described in [Reference Buchstaber, Erokhovets, Masuda, Panov and Park5, Example 3.4] are neither B-rigid nor C-rigid.
Although it follows from Theorem 3.2 and Lemma 4.1 that any product of B-rigid polytopes is C-rigid, it is not clear whether the class of C-rigid polytopes is closed under products. In this direction, we have the following result.
Theorem 4.3 Let M be a quasitoric manifold over a simple polytope P. If P is indecomposable, then
$H^\ast (M;k)$
cannot be decomposed as a nontrivial tensor product for any field k, and, in particular, M is indecomposable up to homotopy.
Proof We first note, with notation as above, that for any basis
$z_1,\ldots ,z_{m-n}$
of
$H^2(M;k)$
there are isomorphisms
$$ \begin{align*} H_*\big(\operatorname{Kos}^{{H^*(M;k)}}(z_1,\ldots,z_{m-n})\big)&\cong H_*\big( \operatorname{Kos}^{{k[P]}}(t_1,\ldots,t_{n},z_1,\ldots,z_{m-n})\big)\\ &\cong H_*\big(\operatorname{Kos}^{{k[P]}}(v_1,\ldots,v_m)\big)\\ & \cong H_\ast(\mathcal{Z}_{P};k); \end{align*} $$
the first isomorphism follows from [Reference Bruns and Herzog3, Corollary 6.1.13 (b)]; the second isomorphism exists because
$t_1,\ldots ,t_{n},z_1,\ldots ,z_{m-n}$
generate the same ideal as
$u_1,\ldots ,u_m$
, and therefore have isomorphic Koszul complexes; and the third isomorphism is Theorem 2.4.
Now suppose that
$H^*(M;k)\cong A_1\otimes A_2$
, with neither
$A_1$
nor
$A_2$
isomorphic to k. Take bases
$z_1,\ldots ,z_i$
of
$A_1^2$
and
$z_{i+1},\ldots ,z_{m-n}$
of
$A_2^2$
. The Künneth isomorphism yields
$$ \begin{align*} H_*\big(\operatorname{Kos}^{A_1}(z_1,\ldots,z_{i})\big)\otimes H_*\big(&\operatorname{Kos}^{A_2}(z_{i+1},\ldots,z_{m-n})\big) \\ & \cong H_*\big( \operatorname{Kos}^{A_1\otimes A_2}(z_1,\ldots,z_{m-n})\big). \end{align*} $$
We may use the chain of isomorphisms above to conclude that this is isomorphic to
$H^\ast (\mathcal {Z}_{P};k)$
, since
$H^2(M;k)\cong A_1^2\oplus A_2^2$
. Thereby, we obtain a tensor product decomposition of
$H^\ast (\mathcal {Z}_{P};k)$
. By Corollary 3.3, the polytope P decomposes accordingly into a product of polytopes.
Remark 4.4 According to Theorem 4.3, if the cohomology
$H^\ast (M)$
of a quasitoric manifold M over P admits a nontrivial tensor product decomposition, then the polytope P decomposes as a product. We note that, unlike the case of moment-angle manifolds, the converse of this result is not true: many quasitoric manifolds over products of polytopes have indecomposable cohomology rings. For instance, the connected sum
$\mathbb {C}P^3 \# \mathbb {C}P^3$
equipped with an appropriate
$T^3$
-action is a quasitoric manifold over the prism
$\Delta ^1\times \Delta ^2$
, while
$H^\ast (\mathbb {C}P^3 \# \mathbb {C}P^3)$
admits no nontrivial tensor product decomposition.
Since a simplex
$\Delta ^n$
is trivially B-rigid, Corollary 3.5 implies that B-rigid polytopes are closed under products with simplices. We next consider a consequence of this fact for quasitoric manifolds and their orbit polytopes.
We first review a common method for constructing new quasitoric manifolds from a given one M. Let
$E=\bigoplus _{i=0}^\ell \xi _i$
be a Whitney sum of complex line bundles over M. Removing the zero section and quotienting by the
$\mathbb {C}^\ast $
-action along each fiber, we obtain the projectivization
$P(E)$
with fiber
$\mathbb {C}P^\ell $
. Starting with
$M=\{ \mathrm {point} \}$
, for example, and iterating this construction yields a tower of projective bundles
$$\begin{align*}B_h \to B_{h-1} \to\cdots \to B_1 \to M=\{ \mathrm{point} \}, \end{align*}$$
called a generalized Bott tower, where
$B_j=P\left (\bigoplus _{i=0}^{\ell _j} \xi _i\right )$
for some complex line bundles
$\xi _0,\ldots ,\xi _{\ell _j}$
over
$B_{j-1}$
. In this case, the generalized Bott manifold
$B_h$
at height h is a quasitoric manifold (in fact, a smooth projective toric variety) over
$\prod _{i=1}^h \Delta ^{\ell _i}$
. Note that since a product of simplices is B-rigid and hence C-rigid, generalized Bott manifolds provide a large class of quasitoric manifolds whose cohomology rings uniquely determine their orbit polytopes.
Let
$G_m$
denote the m-gon,
$m\geqslant 3$
. In [Reference Choi and Park11], Choi and Park prove that the product of a simplex and a polygon is B-rigid and use this to obtain the following result. Following [Reference Choi and Park11], we call
$P(E)$
a projective bundle only when the vector bundle E is a Whitney sum of complex line bundles, as above.
Theorem 4.5 [Reference Choi and Park11, Corollary 4.1]
Let
$P(E)$
be a projective bundle over a
$4$
-dimensional quasitoric manifold. If the cohomology ring of a quasitoric manifold M is isomorphic to that of
$P(E)$
, then the orbit space of M is combinatorially equivalent to
$\Delta ^\ell \times G_m$
.
Using Corollary 3.5, we can extend the result above in two ways. First, since a
$4$
-dimensional quasitoric manifold is precisely a quasitoric manifold over a (B-rigid) polygon
$G_m$
, we can generalize to higher dimensions by replacing the role of the polygon with a B-rigid polytope Q of arbitrary dimension. Second, we can consider towers of iterated projective bundles starting with a quasitoric manifold over Q since a B-rigid polytope remains so after taking a product with a simplex any number of times.
We will need the following lemma, which is well known for smooth projective toric varieties.
Lemma 4.6 Let M be a quasitoric manifold over a simple polytope Q. If
$P(E)$
is a projective bundle over M, then
$P(E)$
is a quasitoric manifold over
$\Delta ^\ell \times Q$
.
Proof Assume
$P(E)$
is a projective bundle over a quasitoric manifold
$M^{2n}$
with
$M^{2n}/T^n=Q$
. Then
$E=\bigoplus _{i=0}^\ell \xi _i$
for some complex line bundles
$\xi _0,\ldots ,\xi _\ell $
over
$M^{2n}$
, so E has a
$T^{\ell +1}$
-action defined by coordinatewise multiplication along each fiber. Moreover, since the fiber inclusion
$\iota \colon M^{2n} \to ET^n \times _{T^n} M^{2n}$
of the Borel construction induces a surjection
$\iota ^\ast \colon k[Q] \to k[Q]/(t_1,\ldots ,t_n)$
, the first Chern class of each line bundle satisfies
$c_1(\xi _i) \in \iota ^\ast (H^2_{T^n}(M^{2n};\mathbb {Z}))$
and it follows from [Reference Mundet i Riera18, Theorem 1.1] that the locally standard
$T^n$
-action on
$M^{2n}$
lifts to a linear action on the total space of each
$\xi _i$
. The resulting action on E makes
$E\to M^{2n}$
a
$T^n$
-equivariant vector bundle, and since this action commutes with the
$T^{\ell +1}$
-action described above, these define an action of
$T^{n+\ell +1}$
on E. The induced
$T^{n+\ell }$
-action on
$P(E)$
is a locally standard half-dimensional torus action giving
$P(E)$
the structure of a quasitoric manifold over
$\Delta ^\ell \times Q$
, since the orbit space
$\mathbb {C}P^\ell /T^\ell $
of the standard action on the fiber can be identified with
$\Delta ^\ell $
, and
$M^{2n}/T^n=Q$
.
Theorem 4.7 Let Q be a B-rigid polytope, and let
$P(E)$
be an iterated projective bundle over any quasitoric manifold over Q. If the cohomology ring of a quasitoric manifold M is isomorphic to that of
$P(E)$
, then the orbit space of M is combinatorially equivalent to
$\prod _{i=1}^h \Delta ^{\ell _i} \times Q$
.
Proof Let
$N^{2n}$
be a quasitoric manifold with
$N^{2n}/T^n=Q$
. If
$P(E)$
is an iterated projective bundle of height h over
$N^{2n}$
, then iteratively applying Lemma 4.6 shows that
$P(E)$
is a quasitoric manifold over a polytope
$\prod _{i=1}^h \Delta ^{\ell _i} \times Q$
. Now if M is a quasitoric manifold over some simple polytope P with
$H^\ast (M;k)\cong H^\ast (P(E);k)$
, then by Lemma 4.1, there is an isomorphism of graded rings
$$\begin{align*}H^\ast(\mathcal{Z}_P;k) \cong H^\ast(\mathcal{Z}_{\prod_{i=1}^h \Delta^{\ell_i} \times Q};k).\end{align*}$$
Since
$\prod _{i=1}^h \Delta ^{\ell _i} \times Q$
is B-rigid by Corollary 3.5, P is combinatorially equivalent to
$\prod _{i=1}^h \Delta ^{\ell _i} \times Q$
.
Acknowledgment
The authors are grateful to the referee for helpful comments which improved the exposition of this paper.
