COHOMOLOGICAL RIGIDITY OF MANIFOLDS DEFINED BY
RIGHT-ANGLED 3-DIMENSIONAL POLYTOPES
VICTOR BUCHSTABER, NIKOLAY EROKHOVETS, MIKIYA MASUDA, TARAS PANOV,AND SEONJEONG PARK
Abstract. We consider the class P of 3-dimensional simple polytopes P
which are flag and do not have 4-belts of facets. It includes fullerenes, i. e.
simple 3-polytopes with only 5-gonal and 6-gonal facets. According to a theorem of Pogorelov, any polytope from P admits a right-angled realisation in
Lobachevsky 3-space, and such a realisation is unique up to isometry.
We study two families of smooth manifolds associated with polytopes fromthe class P. The first family consists of 6-dimensional quasitoric manifolds overpolytopes from P. The second family consists of 3-dimensional small coversof polytopes from P or, equivalently, hyperbolic 3-manifolds of Löbell type.
Our main result is that both families are cohomologically rigid, i. e. two manifolds M and M 0 from either of the families are diffeomorphic if and only iftheir cohomology rings are isomorphic. We also prove that if M and M 0 are
diffeomorphic, then their corresponding polytopes P and P 0 are combinatorially equivalent. These results are intertwined with the classical subjects ofgeometry and topology, such as combinatorics of 3-polytopes, the Four Colour
Theorem, diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.

Contents
1. Introduction
Acknowledgements
2. Preliminaries
2.1. Simple polytopes
2.2. Toric varieties and manifolds
2.3. Quasitoric manifolds
2.4. Small covers and hyperbolic 3-manifolds.
2.5. Topological toric manifolds
2.6. Simplicial complexes and face rings
2.7. Moment-angle complexes and manifolds
2.8. Cohomology of moment-angle complexes
2.9. Moment-angle manifolds and quasitoric manifolds
3. Cohomological rigidity
4. The Pogorelov class: flag 3-polytopes without 4-belts
5. Main results
6. Classification of 6-dimensional manifolds and related problems
Appendix A. Belts in 3-polytopes
Appendix B. Proof of Lemma 4.6.
Appendix C. Proof of Lemma 4.11
References

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The research of the first, second and fourth author was supported by the Russian Foundationfor Basic Research (grants RFBR 14-01-00537 and 16-51-55017-GFEN). The second author was
supported by Young Russian Mathematics award. The third author was supported by JSPS Grantin-Aid for Scientific Research (C) 16K05152. The fourth author was supported by a grant from
Dmitri Zimin’s ‘Dynasty’ foundation.
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V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

1. Introduction
There is the following naive question from the early days of differential topology: given two closed smooth manifolds M and M 0 , when does an isomorphism
H ∗ (M ) ∼= H ∗ (M 0 ) of integral cohomology rings imply that M and M 0 are diffeomorphic? This is generally regarded as an unlikely case, as in the 20th century topologists discovered many important series of manifolds for which the cohomology ring,or even the homotopy type, does not determine the diffeomorphism class. Threedimensional lens spaces, Milnor’s exotic spheres and Donaldson’s four-dimensional
manifolds are prominent examples of different level of complexity.
We say that a family of closed smooth manifolds is cohomologically rigid if a
cohomology ring isomorphism H ∗ (M ) ∼= H ∗ (M 0 ) implies a diffeomorphism M ∼=0
M for any two manifolds in the family.
In this paper we establish cohomological rigidity for two particular families ofmanifolds of dimension 3 and 6, respectively. Each of these families arises froma particularly important class of combinatorial polytopes, which we refer to as
the Pogorelov class P. It consists of simple 3-dimensional polytopes which areflag and do not have 4-belts. In particular, polytopes in P do not have triangular and quadrangular facets. The class P includes combinatorial fullerenes, i. e.
simple 3-polytopes with only pentagonal and hexagonal facets. By the results ofPogorelov [40] and Andreev [1], the class P consists precisely of combinatorial 3polytopes which can be realised in Lobachevsky space L3 with right angles between
adjacent facets (right-angled 3-polytopes for short). The conditions specifying the
Pogorelov class also feature as the “no 4-condition” and “no -condition” in Gromov’s theory of hyperbolic groups [29].
Our first family consists of hyperbolic 3-manifolds of Löbell type. These manifolds
were introduced and studied by Vesnin [43]. They arise from right-angled realisations of polytopes from the Pogorelov class P (see the details in Subsection 2.4).
Each hyperbolic 3-manifold N of Löbell type is composed of 8 copies of a polytope
P ∈ P and is a small cover of P in the sense of Davis and Januszkiewicz [22]. We
prove (in Theorem 5.4) that two such manifolds N and N 0 are diffeomorphic if andonly if their Z2 -cohomology rings are isomorphic.
Our second family arises from toric topology: it consists of quasitoric (or topological toric) manifolds whose quotient polytopes are in the class P. These are
6-dimensional smooth manifolds acted on by a 3-torus T 3 with quotient P ∈ P. We
show (in Theorem 5.2 and Corollary 5.3) that this family is cohomologically rigid,
i. e. two manifolds M and M 0 in the family are diffeomorphic if and only if theircohomology rings are isomorphic. In general a diffeomorphism between quasitoric
manifolds M and M 0 does not imply that the corresponding polytopes P and P 0
are combinatorially equivalent, but this is the case when the quotient polytopes arein the class P (see Theorem 5.2).
Our proofs use both combinatorial and cohomological techniques of toric topology, and build upon recent important results of Fan, Ma and Wang on cohomologicalrigidity of moment-angle manifolds [27], [28]. These authors come to the class P
independently, by considering the property of B-rigidity for simple polytopes (seeSection 3). The class P also featured in the work of Buchstaber and Erokhovets [8]on combinatorial classification of fullerenes.
We note that cohomological rigidity is open for the whole family of toric or topological toric manifolds. In fact, we find it quite surprising that no counterexamples
to the “toric cohomological rigidity problem” were found up to date.
In real dimension 6 the families of quasitoric and topological toric manifolds
coincide and contain strictly the family of toric manifolds (smooth complete toricvarieties). The family of quasitoric (or topological toric) manifolds whose quotient

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

3

polytopes are in the class P is large enough, as there is at least one quasitoric
manifold over any simple 3-polytope. Indeed, the Four Colour Theorem implies
that any simple 3-polytope admits a “characteristic function” (see Proposition 2.8);
this remarkable observation was made by Davis and Januszkiewicz in [22]. Toric
manifolds whose associated polytopes are in P are fewer, but still abundant; many
concrete examples were produced recently by Suyama [42]. On the other hand, there
are no projective toric manifolds whose associated polytopes are in P. This follows
from a result of C. Delaunay [23] that a Delzant 3-polytope must have at least one
triangular or quadrangular face.
Our results on cohomological rigidity of toric manifolds chime with the problem
of diffeomorphism classification for simply connected oriented manifolds, which isa classical subject of algebraic and differential topology. The foundations of thisclassification in dimensions > 5 were laid in the works of Browder and Novikov(see [5], [39]). Novikov [38] showed that for a given simply connected oriented manifold M of dimension > 5 there are only finitely many manifolds M 0 for which there'
exists a homotopy equivalence M −→ M 0 preserving the Pontryagin classes. Important classification results in dimension 6 were obtained in the works of Wall [44],Jupp [32] and Zhubr [45].
Toric, quasitoric or topological toric manifolds M are simply connected, andtheir cohomology rings H ∗ (M ) are generated by 2-dimensional classes. According
to the general results mentioned above, two such manifolds in dimension 6 are diffeomorphic if there is an isomorphism of their cohomology rings preserving the first
Pontryagin class p1 . Therefore, the toric cohomological rigidity problem in dimension 6 reduces to establishing the invariance of p1 under cohomology isomorphisms.
This turns out to be a purely combinatorial and linear algebra problem, see details
in Section 6. However, we were not able to prove directly the invariance of p1 under cohomology isomorphisms for toric manifolds over simple 3-polytopes from theclass P. One of our main results (Theorem 5.2) can be interpreted as a classification
result for a particular family of simply connected 6-dimensional manifolds, and its
proof is independent of the general classification results of Wall, Jupp and Zhubr.

Acknowledgements. The authors are grateful to– Feifei Fan for introducing us to the results of the works [27] and [28] on thecohomological rigidity of moment-angle complexes;– Alexandr Gaifullin for drawing our attention to Siebennman’s no condition in Gromov’s theory of hyperbolic groups, which is equivalent tothe absence of 4-belts in the case of polytopes;– Sergei Novikov for stimulating discussions on the diffeomorphism classification of simply connected manifolds and invariance of Pontryagin classes;– Andrei Vesnin for his help in linking our results to hyperbolic geometry;in particular, to hyperbolic 3-manifolds of Löbell type and the theorem ofPogorelov on right-angled 3-polytopes.
We thank the Institute for Mathematical Science, National University of Singapore, and the organisers of the Program on Combinatorial and Toric Homotopy,
where we were able to exchange ideas which laid the foundation of this work. We
also thank the JSPS program for Advancing Strategic International Networks to
Accelerate the Circulation of Talented Researchers, based on Osaka City University
Advanced Mathematical Institute (OCAMI), which gave us an opportunity to meet
and work together on this project.

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V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

2. Preliminaries
Here we collect the necessary information about toric varieties, quasitoric manifolds and moment-angle manifolds; the details can be found in [11].
2.1. Simple polytopes. Let Rn be an n-dimensional Euclidean space with thescalar product h , i. A convex polytope P is a nonempty bounded intersection offinitely many half-spaces in some Rn :

(2.1)
P = x ∈ Rn : ha i , x i + bi > 0 for i = 1, . . . , m ,where a i ∈ Rn and bi ∈ R. We often fix a presentation by inequalities (2.1) alongsidewith the polytope P . We assume that P is n-dimensional, that is, the dimensionof the affine hull of P is n. We also assume that each inequality ha i , x i + bi > 0
in (2.1) is not redundant, that is, cannot be removed without changing P . Then P
has m facets F1 , . . . , Fm , whereFi = {x ∈ P : ha i , x i + bi = 0}.
Each facet is a polytope of dimension n − 1. A face of P is a nonempty intersectionof facets. Zero-dimensional faces are vertices, and one-dimensional faces are edges.
We refer to n-dimensional polytopes simply as n-polytopes.
Two polytopes P and Q are combinatorially equivalent (P ' Q) if there isa bijection between their faces preserving the inclusion relation. A combinatorial
polytope is a class of combinatorially equivalent polytopes.
A polytope P is simple if exactly n facets meet at each vertex of P . A simple
polytope P is called a flag polytope if every collection of its pairwise intersecting
facets has a nonempty intersection. An n-simplex ∆n is not flag for n > 2. An
n-cube I n is flag for any n.
We denote by GP the vertex-edge graph of a polytope P , and refer to it simply
as the graph of P . A graph is simple if it has no loops and multiple edges. A
connected graph G is 3-connected if it has at least 6 edges and deletion of any one
or two vertices with all incident edges leaves G connected. The following classicalresult describes the graphs of 3-polytopes.
Theorem 2.1 (Steinitz, see [46, Theorem 4.1]). A graph G is the graph of a 3polytope if and only it is simple, planar and 3-connected.
For k > 4, a k-belt in a simple polytope P is a cyclic sequence Bk = (Fi1 , . . . , Fik )of facets in which only two consecutive facets (including Fik , Fi1 ) have nonempty
intersection. For a 3-belt we assume additionally that Fi1 ∩ Fi2 ∩ Fi3 = ∅.
A 3-polytope P with a triangular facet has a 3-belt around it, unless P = ∆3 .
A simple 3-polytope P 6= ∆3 is flag if and only if it does not contain 3-belts.
A fullerene is a simple 3-polytope with only pentagonal and hexagonal facets. A
simple calculation with Euler characteristic shows that the number of pentagonal
facets in a fullerene is 12. The number of hexagonal facets can be arbitrary except
for 1 (see [25, Proposition 2]). Also, any fullerene is a flag polytope without 4-belts(see [26] and [8, Corollary 3.16]).
2.2. Toric varieties and manifolds. A toric variety is a normal complex algebraic variety V containing an algebraic torus (C× )n as a Zariski open subset in sucha way that the natural action of (C× )n on itself extends to an action on V . We only
consider nonsingular complete (compact in the usual topology) toric varieties, also
known as toric manifolds.
There is a bijective correspondence between the isomorphism classes of complex
n-dimensional toric manifolds and complete nonsingular fans in Rn . A fan is a finite
collection Σ = {σ1 , . . . , σs } of strongly convex cones σi in Rn such that every faceof a cone in Σ belongs to Σ and the intersection of any two cones in Σ is a face ofCOHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

5each. A fan Σ is nonsingular (or regular ) if each of its cones σj is generated by partof a basis of the lattice Zn ⊂ Rn (we choose the standard lattice for simplicity).
In particular, each one-dimensional cone of Σ is generated by a primitive vectora i ∈ Zn . A fan Σ is complete if the union of its cones is the whole Rn .
Projective toric varieties are particularly important. A projective toric manifold
V is defined by a lattice Delzant polytope P . Given a simple n-polytope P with vertices in the lattice Zn , the normal fan ΣP has n-dimensional cones σv corresponding
to the vertices v of P , where σv is generated by the primitive inside-pointing normals to the facets of P meeting at v. The polytope P is Delzant whenever its normal
fan ΣP is nonsingular. The fan ΣP defines a projective toric manifold VP . Different
lattice Delzant polytopes with the same normal fan produce different projective
embeddings of the same toric manifold.
Irreducible torus-invariant subvarieties of complex codimension one in V correspond to one-dimensional cones of Σ. When V is projective, they also correspondto the facets of P . We assume that there are m one-dimensional cones (or facets),
denote the corresponding primitive vectors by a 1 , . . . , a m , and denote the corresponding codimension-1 subvarieties by V1 , . . . , Vm .
Theorem 2.2 (Danilov–Jurkiewicz, see [11, Theorem 5.3.1]). Let V be a toric manifold of complex dimension n, with the corresponding complete nonsingular fan Σ.
The cohomology ring H ∗ (V ; Z) is generated by the degree-two classes [vi ] dual tothe invariant submanifolds Vi , and is given byH ∗ (V ; Z) ∼deg vi = 2,
= Z[v1 , . . . , vm ]/I,
where I is the ideal generated by elements of the following two types:(a) vi1 · · · vik such that ai1 , . . . , aik do not span a cone of Σ;mX(b)hai , xivi , for any vector x ∈ Zn .
i=1

It is convenient to consider the integer n × m-matrix
a11 · · · a1m

.. 
..
(2.2)
Λ =  ...
.
. 
an1

···

anm

whose columns are the vectors a i written in the standard basis of Zn . Then theideal (b) of Theorem 2.2 is generated by the n linear forms aj1 v1 + · · · + ajm vm
corresponding to the rows of Λ.
The quotient of a projective toric manifold VP by the action of the compact
torus T n ⊂ (C× )n is the polytope P . When a toric manifold V is not projective, thequotient V /T n has a face structure of a manifold with corners. This face structure
locally looks like that of a simple convex polytope, but globally may fail to be so evencombinatorially. In the case n = 3, however, the quotient V /T 3 is combinatoriallyequivalent to a simple 3-polytope, by Steinitz’s theorem (Theorem 2.1).
2.3. Quasitoric manifolds. In their 1991 work [22] Davis and Januszkiewicz suggested a topological generalisation of projective toric manifolds, which became
known as quasitoric manifolds.
A quasitoric manifold over a combinatorial simple n-polytope P is a topologicalmanifold M of dimension 2n with a locally standard action of T n and a projection
π : M → P whose fibres are the orbits of the T n -action. (An action of T n on M islocally standard if every point x ∈ M is contained in a T n -invariant neighbourhoodequivariantly homeomorphic to an open subset in Cn with the standard coordinatewise action of T n twisted by an automorphism of the torus. The orbit space of a

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V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

locally standard action is a manifold with corners. For a quasitoric manifold M ,the orbit space M/T n is homeomorphic to P .)Not every simple polytope can be the quotient of a quasitoric manifold. Nevertheless, quasitoric manifolds constitute a much larger family than projective toric
manifolds, and enjoy more flexibility for topological applications.
Let F = {F1 , . . . , Fm } be the set of facets of P . Each Mi = π −1 (Fi ) is a quasitoric submanifold of M of codimension 2, called a characteristic submanifold. The
characteristic submanifolds Mi ⊂ M are analogues of the invariant divisors Vi on atoric manifold V . Each Mi is fixed pointwise by a closed one-dimensional subgroup(a subcircle) Ti ⊂ T n and therefore corresponds to a primitive vector λi ∈ Zn defined up to a sign. Choosing a direction of λi is equivalent to choosing an orientation
for the normal bundle ν(Mi ⊂ M ) or, equivalently, choosing an orientation for Mi ,provided that M itself is oriented. An omniorientation of a quasitoric manifold M
consists of a choice of orientation for M and each characteristic submanifold Mi .
The vectors λi are analogues of the generators a i of the one-dimensional conesin the fan corresponding to a toric manifold V , or analogues of the normal vectors
to the facets of P when V is projective. However, the vectors λi need not be thenormal vectors to the facets of P in general.
There is an analogue of Theorem 2.2 for quasitoric manifolds:
Theorem 2.3 ([22]). Let M be an omnioriented quasitoric manifold of dimension 2n over a simple n-polytope P . The cohomology ring H ∗ (M ; Z) is generated
by the degree-two classes [vi ] dual to the oriented characteristic submanifolds Mi ,and is given byH ∗ (M ; Z) ∼deg vi = 2,
= Z[v1 , . . . , vm ]/I,
where I is the ideal generated by elements of the following two types:(a) vi1 · · · vik such that Fi1 ∩ · · · ∩ Fik = ∅ in P ;mX
hλi , xivi , for any vector x ∈ Zn .
(b)i=1

We record a simple corollary for the latter use.
Corollary 2.4. In the notation of Theorem 2.3,(a) the product [vi1 ] · · · [vin ] of n different classes is a generator of H 2n (M ) ∼=Z
if Fi1 ∩ · · · ∩ Fin 6= ∅ and is zero otherwise;(b) for i 6= j, we have [vi ][vj ] = 0 if and only if Fi ∩ Fj = ∅.
By analogy with (2.2), we consider the integer characteristic matrix
λ11 · · · λ1m

.. 
..
(2.3)
Λ =  ...
.
. 
λn1

···

λnm

whose columns are the vectors λi written in the standard basis of Zn . The matrix
Λ has the following property:(2.4)

det(λi1 , . . . , λin ) = ±1

whenever Fi1 ∩ · · · ∩ Fin 6= ∅ in P .

Note that the ideal (b) of Theorem 2.3 is generated by the n linear forms λj1 v1 +· · · + λjm vm corresponding to the rows of Λ.
A map λ : F → Zn , Fi 7→ λi , satisfying (2.4) is called a characteristic function
for a simple n-polytope P . One can produce a characteristic matrix Λ from acharacteristic function λ by fixing an order of facets. A characteristic pair (P, Λ)consists of a simple polytope P and its characteristic matrix Λ.

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES7

A quasitoric manifold M defines a characteristic pair (P, Λ). On the other hand,each characteristic pair gives rise to a quasitoric manifold as follows.
Construction 2.5 ([22]). Let (P, Λ) be a characteristic pair. For each facet Fi of
P we denote by Ti the circle subgroup of T n = Rn /Zn corresponding to the ith
column λi ∈ Zn of the characteristic matrix Λ. For each point x ∈ P , define a torusYTi ,T (x) =
i : x∈Fi

assuming that T (x) = {1} if there are no facets containing x. Property (2.4) implies
that T (x) embeds as a subgroup in T n . Then define
M (P, Λ) = P × T n / ∼,where the equivalence relation ∼ is given by (x, t) ∼ (x0 , t0 ) whenever x = x0 andt0 − t ∈ T (x). One can see that M (P, Λ) is a quasitoric manifold over P .
Change of basis in the lattice results in multiplying Λ from the left by a matrix
from GL (n, Z). Changing the orientation of the ith characteristic submanifold Mi
in the omniorientation data results in changing the sign of the ith column of Λ. A
combinatorial equivalence between polytopes P and P 0 allows us to identify theirsets of facets F and F 0 and therefore identify their characteristic functions. Theseobservations lead us to the following definition.
Definition 2.6. Two characteristic pairs (P, Λ) and (P 0 , Λ0 ) are equivalent if(a) P and P 0 are combinatorially equivalent, and(b) Λ0 = AΛB, where A ∈ GL (n, Z) and B is a diagonal (m × m)-matrix with
±1 on the diagonal.
Quasitoric manifolds M (P, Λ) and M (P 0 , Λ0 ) corresponding to equivalent pairs
are equivariantly homeomorphic (in the weak sense). The latter means that there∼=
is a homeomorphism f : M (P, Λ) −→ M (P 0 , Λ0 ) satisfying f (t · x) = ψ(t) · f (x) forn
any t ∈ T and x ∈ M (P, Λ), where ψ : T n → T n is the automorphism of the torus
given by the matrix A. Furthermore, we have
Proposition 2.7 ([22, Proposition 1.8] and [11, Proposition 7.3.8]). There is a oneto-one correspondence between equivariant homeomorphism classes of quasitoric
manifolds and equivalence classes of characteristic pairs. In particular, for any quasitoric manifold M over P with characteristic matrix Λ, there is an equivariant
homeomorphism M ∼= M (P, Λ).
Remark. Both M and M (P, Λ) were defined as topological manifolds in [22]. The
manifold M (P, Λ) can be endowed with a canonical smooth structure by defining
it as the quotient of the moment-angle manifold ZP by a smooth torus action,see [12] and Subsection 2.9. Nevertheless, for a smooth quasitoric manifold M , theexistence of a diffeomorphism M ∼= M (P, Λ) is a delicate issue, see the discussionin [11, §7.3]. On the other hand, in the case of 6-dimensional quasitoric manifolds(which is our main concern in this paper), such a diffeomorphism follows from theclassification results of Wall and Jupp discussed in Section 6.
In dimensions n > 4, there are simple n-polytopes P which do not admit any
characteristic matrix Λ, see [22, 1.22]. Such a polytope cannot be the quotient of aquasitoric manifold. On the other hand, we have the following observation by Davisand Januszkiewicz, whose proof remarkably uses the Four Colour Theorem:
Proposition 2.8 ([22]). Any simple 3-polytope admits a characteristic matrix Λ.

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V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

Proof. Given a 4-colouring of the facets of P , we assign to a facet of ith colour
the ith basis vector e i ∈ Z3 for i = 1, 2, 3 and the vector e 1 + e 2 + e 3 for i =4. The resulting 3 × m-matrix Λ satisfies (2.4), as any three of the four vectors
e 1 , e 2 , e 3 , e 1 + e 2 + e 3 form a basis of Z3 .

A projective toric manifold is a quasitoric manifold. A non-projective toric manifold V may fail to be quasitoric, as the quotient manifold with corners V /T n is
not necessarily a simple polytope, even combinatorially. First examples of this sortappear in dimension n = 4, see [41]. All complex 3-dimensional toric manifolds,
even non-projective ones, are quasitoric by the Steinitz theorem (Theorem 2.1).
2.4. Small covers and hyperbolic 3-manifolds. Replacing the torus T n in thedefinition of a quasitoric manifold by the subgroup Zn2 ⊂ T n (n commuting involutions), one obtains the definition of a small cover [22]. A small cover of a simple
n-polytope P is a manifold N of dimension n with a locally standard action of Zn2
and a projection π : N → P whose fibres are the orbits of the Zn2 -action.
The set of real points of a projective toric manifold VP (i. e. the set of points
fixed under complex conjugation) is a small cover of P ; it is sometimes called a real
toric manifold.
The theory of small covers parallels that of quasitoric manifolds, and we justoutline the most crucial points.
Theorem 2.9 ([22]). Let N be a small cover of a simple n-polytope P . The cohomology ring H ∗ (N ; Z2 ) is generated by the degree-one classes [vi ] dual to thecharacteristic submanifolds Ni , and is given by∼ Z2 [v1 , . . . , vm ]/I,H ∗ (N ; Z2 ) =deg vi = 1,
where I is the ideal generated by elements of the following two types:(a) vi1 · · · vik such that Fi1 ∩ · · · ∩ Fik = ∅ in P ;mX
hλi , xivi , for any vector x ∈ (Z2 )n .
(b)
i=1

The characteristic matrix Λ corresponding to a small cover N has entries in Z2
and satisfies the same condition (2.4). The equivalence of Z2 -characteristic pairs
is defined in the same way as in the quasitoric case, with GL(n, Z) replaced by
GL(n, Z2 ). A small cover N of P is equivariantly homeomorphic to the “canonicalmodel”N (P, Λ) = P × Zn2 / ∼with the equivalence relation ∼ defined as in the quasitoric case. Note that N (P, Λ)is composed of 2n copies of the polytope P , patched together along their facets.
Reducing a Z-characteristic matrix mod 2 we obtain a Z2 -characteristic matrix.
The following question is open.
Problem 1. Assume given a Z2 -characteristic pair (P, Λ) consisting of a simple
n-polytope P and an (n × m)-matrix Λ with entries in Z2 satisfying (2.4). CanΛ be obtained by reduction mod 2 from an integer matrix satisfying the same
condition (2.4)?
The answer to the above problem is positive for 3-polytopes:
Proposition 2.10. Every Z2 -characteristic pair (P, Λ) with 3-dimensional P is
the mod 2 reduction of a Z-characteristic pair.
Proof. One can check that any (3 × 3)-matrix with entries 0 or 1 and determinant 1
mod 2 has determinant ±1 when viewed as an integer matrix. Ineed, such a matrix

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

9



1 0 1
either has a column with 2 zeros, or is 1 1 1 up to permutation of rows and0 1 1
columns. The required property is then verified directly.

A particularly important class of 3-dimensional small covers are hyperbolic 3manifolds of Löbell type, introduced and studied by Vesnin in [43].
Construction 2.11. Let P be a polytope in 3-dimensional Lobachevsky space L3
with right angles between adjacent facets (a right-angled 3-polytope for short). Denote by G(P ) the group generated by reflections in the facets of P . It is a right-angled
Coxeter group given by the presentation
G(P ) = hg1 , . . . , gm | gi2 = 1, gi gj = gj gi if Fi ∩ Fj 6= ∅i,where gi denotes the reflection in the facet Fi . The reflections in adjacent facets
commute because of the right-angledness. There are no relations between the reflections in non-adjacent faces, as the corresponding reflection hyperplanes do notintersect in L3 . The group G(P ) acts on L3 discretely with finite isotropy subgroups
and with fundamental domain P .
Now consider an epimorphism ϕ : G(P ) → Z32 . As was observed in [43, Lemma 1],
the subgroup Ker ϕ ⊂ G(P ) does not contain elements of finite order if and only if
the images of the reflections in any three facets of P that have a common vertex
are linearly independent in Z32 . In this case the group Ker ϕ acts freely on L3 .
The quotient N = L3 / Ker ϕ is called a hyperbolic 3-manifold of Löbell type. It is
composed of |Z32 | = 8 copies of P and has a Riemannian metric of constant negative
curvature. Furthermore, such a manifold N is aspherical (has the homotopy typeof Eilenberg–Mac Lane space K(G(P ), 1)), as its universal cover L3 is contractible.
ab

Since the abelianisation of G(P ) is Zm
2 , the epimorphism ϕ factors as G(P ) −→m Λ
Z2 −→ Z32 , where Λ is a linear map. The above condition for the freeness of theaction of Ker ϕ on L3 is equivalent to that Λ satisfies (2.4) (i. e. Λ is given by aZ2 -characteristic matrix). We therefore can identify N = L3 / Ker ϕ with the small
cover N (P, Λ).
Which combinatorial 3-polytopes have right-angled realisations in L3 ? Pogorelov
stated necessary conditions in 1967, which were later shown by Andreev to be
sufficient:
Theorem 2.12 ([40], [1]). A combinatorial 3-polytope can be realised as a rightangled polytope in Lobachevsky space L3 if and only if it is simple, flag and does
not have 4-belts. Furthermore, such a realisation is unique up to isometry.
We refer to the class of simple flag 3-polytopes without 4-belts as the Pogorelov
class P. It will feature prominently throughout the rest of our paper.
A polytope from the class P does not have triangular or quadrangular facets.
According to a result of Došlić [26] (see also [7, Corollary 3.16]), the Pogorelov classcontains all fullerenes.
We summarise the constructions and results above as follows.
Theorem 2.13. A small cover N (P, Λ) of a 3-polytope P from the Pogorelov
class P has the structure of a hyperbolic 3-manifold of Löbell type L3 / Ker ϕ, withab

Λ3
the epimorphsim ϕ given by the composition G(P ) −→ Zm
2 −→ Z2 . Furthermore,such a 3-manifold N (P, Λ) is aspherical.

Remark. The conditions specifying the Pogorelov class P also feature in Gromov’s
theory of hyperbolic groups. Namely, the “no 4-condition” from [29, §4.2.E] for a
simplicial complex K is the absence of missing 2-faces, while the “no -condition” is10

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

the absence of chordless 4-cycles. When K is the dual complex of a simple polytope,
these two conditions translate to the absence of 3- and 4-belts, respectively.
The relationship between small covers and hyperbolic manifolds was also mentioned in the work of Davis and Januszkiewicz [22, p. 428], although the criterion
for right-angled realisation of a polytope was stated there incorrectly.
2.5. Topological toric manifolds. A toric manifold is not necessarily a quasitoric
manifold and a quasitoric manifold is also not necessarily a toric manifold. However, both toric and quasitoric manifolds are examples of topological toric manifolds
introduced in [31]. Recall that a toric manifold admits an algebraic action of (C× )nwith an open dense orbit. It has local charts equivariantly isomorphic to a sumof complex one-dimensional algebraic representations of (C× )n . A topological toric
manifold is a compact smooth 2n-dimensional manifold with an effective smoothaction of (C× )n having an open dense orbit and covered by finitely many invariant
open subsets each equivariantly diffeomorphic to a sum of complex one-dimensionalsmooth representation spaces of (C× )n . (The latter condition automatically follows
from the existence of a dense orbit in the algebraic category, but not in the smoothcategory.)
The cohomology ring of a topological toric manifold is described similarly tothe toric or quasitoric case; there is an analogue of Theorems 2.2 or 2.3, see [31,Proposition 8.3].
2.6. Simplicial complexes and face rings. Let K be an (abstract) simplicial
complex on the set [m] = {1, . . . , m}, i. e. K is a collection of subsets I ⊂ [m]such that for any I ∈ K all subsets of I also belong to K. We always assume thatthe empty set ∅ and all one-element subsets {i} ⊂ [m] belong to K. We refer toI ∈ K as a simplex (or a face) of K. Every abstract simplicial complex K has ageometric realisation |K|, which is a polyhedron in a Euclidean space (a union of
convex geometric simplices).
A non-face of K is a subset I ⊂ [m] such that I ∈/ K. A missing face (or aminimal non-face) of K is an inclusion-minimal non-face of K, that is, a subset
I ⊂ [m] such that I is not a simplex of K, but every proper subset of I is a simplexof K. A simplicial complex K is called a flag complex if each of its missing faces
consists of two vertices. Equivalently, K is flag if any set of vertices of K which are
pairwise connected by edges spans a simplex. Every flag complex K is determinedby its 1-skeleton K1 , and is obtained from the graph K1 by filling in all complete
subgraphs by simplices.
Let P be a simple n-polytope with m facets F1 , . . . , Fm . Then
KP = I = {i1 , . . . , ik } ∈ [m] : Fi1 ∩ · · · ∩ Fik 6= ∅is a simplicial complex on [m], called the dual complex of P . The vertices of KP
correspond to the facets of P , and the empty simplex ∅ corresponds to P itself.
Geometrically, |KP | is an (n − 1)-dimensional sphere simplicially subdivided as theboundary of the dual polytope of P .
The definitions of flag polytopes and complexes agree: P is a flag polytope if andonly if KP is a flag complex. A k-belt in P with k > 4 corresponds to a chordlessk-cycle in the graph KP1 .
We fix a commutative ring k with unit.
The face ring of K (also known as the Stanley–Reisner ring) is defined as thequotient of the polynomial algebra k[v1 , . . . , vm ] by the square-free monomial ideal
generated by non-simplices of K:k[K] = k[v1 , . . . , vm ] vi1 · · · vik : {i1 , . . . , ik } ∈/K .

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES11

As k[K] is the quotient of the polynomial ring by a monomial ideal, it has a gradingor even a multigrading (a Zm -grading). We use an even grading: deg vi = 2 andmdeg vi = 2e i , where e i ∈ Zm is the ith standard basis vector.
Note that when K = KP for a simple polytope P , the ring Z[P ] coincides with
the quotient of Z[v1 , . . . , vm ] by the relations (a) in Theorem 2.2 or in Theorem 2.3.
A simplicial complex K is flag if and only if its face ring k[K] is a quadratic algebra, i. e. the quotient of k[v1 , . . . , vm ] by an ideal generated by quadratic monomials(which have degree 4 in our grading).
2.7. Moment-angle complexes and manifolds. Let K be a simplicial complex
on the set [m], and let (D2 , S 1 ) denote the pair of a disc and its boundary circle.
For each simplex I = {i1 , . . . , ik } ∈ K, set(D2 , S 1 )I = {(x1 , . . . , xm ) ∈ (D2 )m : xi ∈ S 1 when i ∈/ I}.
The moment-angle complex is defined as[(2.5)
ZK = (D2 , S 1 )K =(D2 , S 1 )I ⊂ (D2 )m .
I∈Kn−1

If |K| is homeomorphic to a sphere S, then ZK is a topological manifold. If
K is the boundary of a convex simplicial polytope or is a starshaped sphere (theunderlying complex of a complete simplicial fan), then ZK has a smooth structure.
In the polytopal case there is an alternative way to define ZK in terms of the dual
simple polytope P . Namely, assume given a presentation of a convex n-dimensional
polytope P by inequalities (2.1). Define the map
iP : Rn → Rm , x 7→ ha 1 , x i + b1 , . . . , ha m , x i + bm ,mso iP (P ) ⊂ Rm
> = {(y1 , . . . , ym ) ∈ R : yi > 0}. Also, define the mapµ : Cm → R m>,(z1 , . . . , zm ) 7→ (|z1 |2 , . . . , |zm |2 ).

Then define the space ZP by the pullback diagram
ZP −−−−→ Cm



µ
y
y
(2.6)
ip

P −−−−→ Rm>
The space ZP can be written as an intersection of (m − n) Hermitian quadrics in
Cm , and this intersection is nondegenerate precisely when the polytope P is simple.
In the latter case, ZP is a smooth (m + n)-dimensional manifold. Furthermore, themanifold ZP is diffeomorphic to the moment-angle complex ZKP . In particular, thediffeomorphism type of ZP depends only on the combinatorial type of P . We shalltherefore not distinguish between ZP and ZKP and refer to it as the moment-angle
manifold corresponding to a simple polytope P . The details of these constructions
can be found in [11, Chapter 6].
The standard coordinatewise action of the m-torus T m on (D2 )m or Cm induces
the canonical T m -action on ZK or ZP .
2.8. Cohomology of moment-angle complexes. We consider (co)homology
with coefficients in k. Denote by Λ[u1 , . . . , um ] the exterior algebra on m generators over k which satisfy the relations u2i = 0 and ui uj = −uj ui .
The Koszul complex (or the Koszul algebra) of the face ring k[K] is defined as
the differential Z ⊕ Zm -graded algebra (Λ[u1 , . . . , um ] ⊗ k[K], d), where(2.7)

mdeg ui = (−1, 2e i ),mdeg vi = (0, 2e i ),dui = vi ,dvi = 0.

Cohomology of (Λ[u1 , . . . , um ] ⊗ k[K], d) is the Tor-algebra Tork[v1 ,...,vm ] (k[K], k).
It also inherits a Z ⊕ Zm -grading.

12

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

Theorem 2.14 ([4], [11, Theorem 4.5.5]). There are isomorphisms of (multi)graded
commutative algebrasH ∗ (ZK ) ∼= Tork[v1 ,...,vm ] k[K], k∼= H Λ[u1 , . . . , um ] ⊗ k[K], d .
The cohomology of ZK therefore acquires a multigrading, with the multigradedand ordinary graded components of H ∗ (ZK ) given byM
H −i,2J (ZK ) = Tor−i,2JH ` (ZK ) =H −i,2J (ZK ),k[v1 ,...,vm ] k[K], k ,−i+2|J|=`mwhere J = (j1 , . . . , jm ) ∈ Z and |J| = j1 + ·· · + jm .
The Koszul algebra (Λ[u1 , . . . , um ]⊗k[K], d is infinitely generated as a k-module.
We define its quotient algebraR∗ (K) = Λ[u1 , . . . , um ] ⊗ k[K] (vi2 = ui vi = 0, 1 6 i 6 m)with the induced multigrading and differential (2.7). Note that R∗ (K) has a finite kbasis. Passing to R∗ (K) does not change the cohomology. This can be proved either
algebraically [11, Lemma 3.2.6] or using the following topological interpretation:
Lemma 2.15 ([11, Lemma 4.5.3]). The algebra R∗ (K) coincides with the cellularcochains of ZK for the appropriate cell structure. In particular, there is an isomorphism of cohomology algebras
H(R∗ (K)) ∼= H ∗ (ZK ).
The multigraded component R−i,2J (K) is zero unless all coordinates of the vector
J ∈ Zm are 0 or 1, and the same is true for the multigraded
H −i,2J (ZK ).
P cohomologym
We can identify subsets J ⊂ [m] with vectorsj∈J e j ∈ Z . Given J ={j1 , . . . , jk } ⊂ [m], we denote by vJ the monomial vj1 · · · vjk ∈ k[v1 , . . . , vm ],and similarly consider exterior monomials uJ = uj1 · · · ujk ∈ Λ[u1 , . . . , um ]. We
also use the notation uJ vI for the monomial uJ ⊗ vI in the Koszul algebraΛ[u1 , . . . , um ] ⊗ k[K]. Then R∗ (K) has a finite k-basis consisting of monomials
uJ vI where J ⊂ [m], I ∈ K and J ∩ I = ∅.
Given J ⊂ [m], define the corresponding full subcomplex of K as
KJ = {I ∈ K : I ⊂ J}.
Consider simplicial cochains C ∗ (KJ ) with coefficients in k. Let αL ∈ C p−1 (KJ ) be
the basis cochain corresponding to an oriented simplex L = (l1 , . . . , lp ) ∈ KJ ; ittakes value 1 on L and vanishes on all other simplices. Define a k-linear map(2.8)

f : C p−1 (KJ ) −→ Rp−|J|,2J (K),αL 7−→ ε(L, J) uJ\L vL ,Q
where ε(L, J) is the sign given by ε(L, J) = j∈L ε(j, J) and ε(j, J) = (−1)r−1 if
j is the rth element of the set J ⊂ [m] written in increasing order.
Theorem 2.16 ([11, Theorem 3.2.9]). The maps (2.8) combine to an isomorphismof cochain complexes C ∗ (KJ ) → R ∗,2J (K) and induce an isomorphism
−i,2J
e |J|−i−1 (KJ ) ∼H= Tork[v1 ,...,vm ] k[K], k ,e k (KJ ) denotes the kth reduced simplicial cohomology group of KJ .
where H
Theorem 2.17 ([11, Theorem 4.5.8]). There are isomorphisms of k-modules
M
e |J|−i−1 (KJ ),e `−|J|−1 (KJ ).
H −i,2J (ZK ) ∼H ` (ZK ) ∼
H
=H
=
J⊂[m]

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

13

These isomorphisms combine to form a ring isomorphismMe ∗ (KJ ),H ∗ (ZK ) ∼
H
=
J⊂[m]

where the ring structure on the right hand side is given by the canonical maps
H k−|I|−1 (KI ) ⊗ H `−|J|−1 (KJ ) → H k+`−|I|−|J|−1 (KI∪J )which are induced by the simplicial maps KI∪J → KI ∗ KJ for I ∩ J = ∅ and zerootherwise.
Proposition 2.18. The 3-dimensional cohomology H 3 (ZK ) is freely generated by
the cohomology classes [ui vj ] = [uj vi ] corresponding to pairs of vertices i, j suchthat {i, j} ∈/ K. If K = KP for a simple polytope P , then these 3-dimensional
cohomology classes correspond to pairs of non-adjacent facets Fi , Fj .
Example 2.19. Let K = 1 r r 2 3 r r 4 be the union of two segments. Thennontrivial integral cohomology groups of ZK are given below together with thecocycles in the algebra R∗ (K) representing generators:
e −1 (∅) ∼H 0 (ZK ) ∼=Z=HM
e 0 (KJ ) ∼H
H 3 (ZK ) ∼
= Z4
=

1
u1 v3 , u1 v4 , u2 v3 , u2 v4|J|=2

H 4 (ZK ) ∼=M

e 0 (KJ ) ∼H= Z4

u1 u2 v3 , u1 u2 v4 , u3 u4 v1 , u3 u4 v2|J|=3

e 0 (K) ∼H 5 (ZK ) ∼=H=Zu1 u2 u4 v3 − u1 u2 u3 v4

Cochains in C 0 (K) are functions on the vertices of K, and cocycles are functions
which are constant on the connected components of K. In our case, the cocyclee 0 (K). It is mapped by (2.8) to the cocycle
α{3} + α{4} represents a generator of Hu1 u2 u4 v3 − u1 u2 u3 v4 representing a generator of H 5 (ZK ).
Moment-angle complexes ZK may have nontrivial triple Massey products of 3dimensional cohomology classes. First examples (found by Baskakov [3]) appear
already for moment-angle manifolds corresponding to 3-polytopes (see also [11,§4.9]). A complete description of the triple Massey product H 3 (ZK ) ⊗ H 3 (ZK ) ⊗H 3 (ZK ) → H 8 (ZK ) is given by the following result of Denham and Suciu:
Theorem 2.20 ([24, Theorem 6.1.1]). The following are equivalent:(a) there exist cohomology classes α, β, γ ∈ H 3 (ZK ) for which the Massey product hα, β, γi is defined and non-trivial;(b) the graph K1 contains an induced subgraph isomorphic to one of the fivegraphs in Figure 1.
sA A
 s As

s s
 




s


s
A
 A
 s As

s s
 




s


s
A
 A
 s As

s s
 



s



s
A
 A
 s As

s s
 



s



Figure 1. Five graphs.

s
A
 A
 s As s ss 14

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

2.9. Moment-angle manifolds and quasitoric manifolds. Let P be a simple
n-polytope with the dual simplicial complex KP . The existence of a characteristic
matrix (2.3) for P is equivalent to a choice of n linear forms(2.9)

tj = λj1 v1 + · · · + λjm vm ,j = 1, . . . , nsuch that Z[KP ] is a finitely generated free module over Z[t1 , . . . , tn ]. Such t1 , . . . , tnform a linear regular sequence in Z[KP ]. This implies that k[KP ] is a Cohen–
Macaulay ring over any k, but the condition is actually stronger, as it assumes
the existence of a linear regular sequence over Z (and hence over any finite field).
Given a characteristic matrix (2.3) (or a linear regular sequence (2.9) in Z[KP ]),one can define the corresponding homomorphism of tori ΛT : T m → T n . Its kernel
Ker ΛT is an (m − n)-dimensional subtorus in T m that acts freely on ZP . The
quotient ZP / Ker ΛT can be identified with the quasitoric manifold M (P, Λ) from
Construction 2.5. As ZP is a smooth intersection of quadrics (2.6) and the actionof torus is smooth, we obtain a canonical smooth structure on M (P, Λ) as in [12].
We say that T n -manifolds M and M 0 are weakly equivariantly diffeomorphic ifthere is a diffeomorphism f : M → M 0 and an automorphism θ : T n → T n such thatf (t · x) = θ(t) · f (x) for any x ∈ M and t ∈ T n . The following result is immediate.
Proposition 2.21. If characteristic pairs (P, Λ) and (P 0 , Λ0 ) are equivalent, thenthe corresponding quasitoric manifolds M (P, Λ) and M (P 0 , Λ0 ) are weakly equivariantly diffeomorphic.
The general homological properties of regular sequences imply yet another description of the cohomology of ZP :Theorem 2.22 ([10, Theorem 4.2.11], [11, Lemma A.3.5]). Let P be a simple
n-polytope with m facets, and assume there exists a linear integral regular sequence (2.9). Denote by J the ideal in Z[v1 , . . . , vm ] generated by t1 , . . . , tn . Then
there is an isomorphism of cohomology ringsH ∗ (ZP ; Z) ∼= TorZ[v ,...,v ]/J Z[KP ]/J , Z .
1mNote that Z[KP ]/J is the cohomology ring of the quasitoric manifold M (P, Λ),
see Theorem 2.3. The theorem above implies that the spectral sequence of theprincipal T m−n -fibration ZP → M (P, Λ) degenerates at the E3 term.
3. Cohomological rigidity
We continue to consider cohomology with coefficients in a commutative ring withunit k. When k is not specified explicitly, we assume k = Z.
Definition 3.1. We say that a family of closed manifolds is cohomologically rigid
over k if manifolds in the family are distinguished up to homeomorphism by theircohomology rings with coefficients in k. That is, a family is cohomologically rigid
if a graded ring isomorphism H ∗ (M1 ; k) ∼= H ∗ (M2 ; k) implies a homeomorphism
M1 ∼= M2 whenever M1 and M2 are in the family.
There is a homotopical and smooth versions of cohomological rigidity, with homeomorphisms replaced by homotopy equivalences and diffeomorphisms, respectively.
In toric topology, cohomological rigidity is studied for (quasi)toric manifolds andmoment-angle manifolds. We refer to [37], [18] and [11, §7.8] for a more detailed
survey of related results and problems. The main question here is as follows.
Problem 2. Let M1 and M2 be two toric manifolds with isomorphic cohomology
rings. Are they homeomorphic? In other words, is the family of toric manifolds
cohomologically rigid? One can ask the same question for quasitoric and topologicaltoric manifolds, and with homeomorphisms replaced by diffeomorphisms.

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES15

The problem is solved positively for some particular families of toric and quasitoric manifolds, such as cohomologically trivial Bott towers (whose corresponding polytopes are combinatorial cubes) [36], Q-cohomologically trivial Bott towers [14], Z2 -cohomologically trivial Bott towers [15], Bott towers of real dimension
up to 8 [13], quasitoric manifolds over a product of two simplices [21] and oversome dual cyclic polytopes [30]. The problem is open for general Bott towers, andfor (quasi)toric manifolds of real dimension 6 (over 3-dimensional polytopes). The
latter case is the subject of this paper: we give a solution for a particular class of
3-polytopes.
There is also a cohomological rigidity problem for real toric objects, such as real
toric manifolds, small covers, and real topological toric manifolds [31], with Z2 cohomology rings. This problem is solved positively for real Bott towers [16], [33],but negatively in some other cases [35].
Cohomological rigidity is also open for moment-angle manifolds, in both graded
and bigraded versions:
Problem 3. Let ZP1 and ZP2 be two moment-angle manifolds whose (bigraded)cohomology rings are isomorphic. Are they homeomorphic? In other words, is thefamily of moment-angle manifolds cohomologically rigid?
A homeomorphism of two quasitoric manifolds over P1 and P2 , or a homeomorphism of moment-angle manifolds ZP1 and ZP2 does not imply that the polytopes
P1 and P2 are combinatorially equivalent, as shown by the next example.
Example 3.2. A vertex truncation operation [11, Construction 1.1.1] can be applied to a simple polytope P to produce a new simple polytope vt(P ) with one
more facet. If one applies this operation iteratively, then the combinatorial type ofthe resulting polytope depends, in general, on the choice and order of truncatedvertices. For example, applying this operation three times to a 3-simplex one canproduce three combinatorially different polytopes Pi , i = 1, 2, 3, with 7 facets each(their dual simplicial polytopes are known as stacked ). The corresponding momentangle manifolds ZPi are diffeomorphic, see [11, §4.6]. The polytopes Pi have Delzant
realisations such that the correponding toric manifolds VPi are obtained from CP 3
by blowing it up three times in three different ways. Each of VPi is diffeomorphic
to a connected sum of 4 copies of CP 3 (see the details in [37]).
One can look for classes of simple polytopes P whose combinatorial type isdetermined by the cohomology ring of any (quasi)toric manifold over P or by thecohomology ring of the moment-angle manifold ZP . This leads to the following two
notions of rigidity for simple polytopes, considered in [37] and [6] respectively.
Definition 3.3. A simple polytope P is said to be C-rigid if any of the two conditions hold:(a) there are no quasitoric manifolds M over P (equivalently, there are no linear
regular sequences (2.9) in Z[KP ]), or(b) whenever there exist a quasitoric manifold M over P and a quasitoric manifold M 0 over another polytope P 0 with a cohomology ring isomorphism
H ∗ (M ) ∼= H ∗ (M 0 ), there is a combinatorial equivalence P ' P 0 .
We say that a property of simple polytopes is C-rigid if for any ring isomorphism
H ∗ (M ) ∼= H ∗ (M 0 ), both P and P 0 either have or do not have the property.
Definition 3.4. A simple polytope P is said to be B-rigid if any cohomology ringisomorphism H ∗ (ZP ) ∼= H ∗ (ZP 0 ) of moment-angle manifolds implies a combinatorial equivalence P ' P 0 .
We say that a property of simple polytopes is B-rigid if for any ring isomorphism
H ∗ (ZP ) ∼= H ∗ (ZP 0 ), both P and P 0 either have or do not have the property.

16

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

According to Example 3.2, a truncated simplex with at least 3 truncations (thedual to a stacked polytope with at least 3 stacks) is neither C-rigid nor B-rigid.
Previously known examples of C-rigid polytopes include products of simplices andtheir single vertex truncations [19], as well as a product of a simplex and a polygon [20]. Also, C-rigidity was determined in [19] for all simple 3-polytopes with 6 9
facets. The following relation between the two notions of rigidity can be extracted
from the results of [19]:Proposition 3.5. If a simple polytope P is B-rigid, then it is C-rigid.
∼=

Proof. Assume that we have a cohomology ring isomorphism ϕ : H ∗ (M ) −→H ∗ (M 0 ) for quasitoric manifolds M over P and M 0 over P 0 . We need to show∼=that it implies a ring isomorphism ψ : H ∗ (ZP ) −→ H ∗ (ZP 0 ), as the latter would
give P ' P 0 by B-rigidity. Let J and J 0 denote the corresponding ideals in Z[KP ]and Z[KP 0 ], respectively, generated by the linear regular sequences (2.9). Then we∼=
have a ring isomorphism ϕ : Z[KP ]/J −→ Z[KP 0 ]/J 0 . We need to show that thisisomorphism gives rise to a ring isomorphism
 ∼

=
(3.1)
TorZ[v1 ,...,vm ]/J Z[KP ]/J , Z −→ TorZ[v1 ,...,vm0 ]/J 0 Z[KP 0 ]/J 0 , Z ,∼=as the latter is nothing but an isomorphism H ∗ (ZP ) −→ H ∗ (ZP 0 ) according to
Theorem 2.22. This is is proved in [19, Lemma 3.7]. Namely, the isomorphism∼=
ϕ : Z[KP ]/J −→ Z[KP 0 ]/J 0 can be extended to a commutative diagram
Z[v1 , . . . , vm ]/J
Z[KP ]/J

∼
=

ϕ
∼
=

/ Z[v1 , . . . , vm0 ]/J 0
/ Z[KP 0 ]/J 0 ,implying in particular that m = m0 . The commutative diagram above gives riseto an isomorphism (3.1) by the standard properties of Tor. More specifically, theisomorphism ϕ gives an isomorphism of the Koszul algebras ∼=(3.2) ϕe : Λ[u1 , . . . , um ]/J ⊗ Z[KP ]/J , d −→ Λ[u01 , . . . , u0m ]/J 0 ⊗ Z[KP 0 ]/J 0 , d ,where the ideals in the exterior algebras are defined by the same linear forms as in
the face rings. Then (3.1) is obtained by passing to the cohomology.

Remark. The argument above is essentially [19, Lemma 3.7]. The term “B-rigidity”
was introduced in the last section of [19]. However, the implication of Proposition 3.5
was erroneously stated there in the opposite direction: “if P is C-rigid, then it is
B-rigid”. This was a confusion. It is not known whether C-rigidity is equivalent to
B-rigidity, and it is unlikely to be true in general.
4. The Pogorelov class: flag 3-polytopes without 4-belts
Recall that the Pogorelov class P consists of simple 3-polytopes P which are flagand do not have 4-belts (or, equivalently, of simple 3-polytopes P 6= ∆3 without 3and 4-belts). In this section we consider combinatorial properties of polytopes P ∈P and cohomological properties of the corresponding moment-angle manifolds ZP .
The key statements here are Theorem 4.8, Theorem 4.10 and Lemma 4.11; they
will be used in the proof of the main results in the next section.
The first property is straightforward:
Proposition 4.1. In a polytope P ∈ P, there are no 3-gonal or 4-gonal facets.

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

17

Lemma 4.2. A simple 3-polytope P is flag if and only if each of its facets F issurrounded by a k-belt with k > 4.
Proof. Note that k is the number of edges in F .
Assume that P is flag. Let (F1 , . . . , Fk ), k > 4, be the sequence of facets adjacent
to F , written in a cyclic order. The claim is that it forms a k-belt. Indeed, otherwise
there is a pair of facets Fi , Fj with Fi ∩ Fj 6= ∅ and j − i 6≡ ±1 mod k. Then eachpair of facets in the triple F, Fi , Fj has nonempty intersection, but F ∩ Fi ∩ Fj = ∅.
This contradicts the assumption that P is flag.
Now assume that P is not flag. Then either P = ∆3 , or P has a 3-belt, and noneof the facets in this 3-belt is surrounded by a belt.
Lemma 4.3. For any two facets Fi and Fj in a polytope P ∈ P, there is a vertexx∈/ Fi ∪ Fj .
Proof. Take any facet F` different from Fi and Fj . Then F` has at most two common
vertices with Fi and at most two common vertices with Fj . On the other hand, F`
has at least 5 vertices by the previous corollary. Thus, at least one vertex of F` does
not lie in Fi ∪ Fj .

Lemma 4.4. In a flag 3-polytope P , for any facet Fi there is a facet Fj such thatFi ∩ Fj = ∅.
Proof. By Lemma 4.2 the facet Fi is surrounded by a k-belt Bk with k > 4. Then∂P \ Bk consists of two connected components: one of them is the interior of Fi ,and the other contains the interior of a facet Fj that we look for.

Now we consider cohomology of moment-angle manifolds ZP with coefficientsin Z. We recall from Proposition 2.18 that H 3 (ZP ) has a basis of cohomology
classes [ui vj ] = [uj vi ] corresponding to pairs of non-adjacent facets Fi , Fj .
Proposition 4.5. Let P be a simple 3-polytope with m facets and let K = KP be
its dual simplicial complex. In the notation of Theorem 2.17, we havee −1 (K∅ ) = Z
for ` = 0,
H

L
L


0
1
e
e
|I|=`−1 H (KI ) ⊕|I|=`−2 H (KI ) for 3 6 ` 6 m,
H ` (ZP ) =2e
H (K) = Zfor ` = m + 3,0otherwise.
In particular, H ∗ (ZP ) does not have torsion. Furthermore, all nontrivial productsin H ∗ (ZP ) are of the forme 0 (KI ) ⊗ H
e 0 (KJ ) → H
e 1 (KI∪J ),H

I ∩ J = ∅,ore 0 (KI ) ⊗ H
e 1 (K[m]\I ) → H
e 2 (K) = Z.
H
For the multigraded components of H ∗ (ZP ), these two cases correspond toH −(|I|−1), 2I (ZP ) ⊗ H −(|J|−1), 2J (ZP ) → H −(|I|+|J|−2), 2(ItJ) (ZP ),H −(|I|−1), 2I (ZP ) ⊗ H −(m−|I|−2), 2([m]\I) (ZP ) → H −(m−3), 2[m] (ZP ) = Z,
where the latter is the Poincaré duality pairing.
Proof. This follows from Theorem 2.14, Theorem 2.16 and Theorem 2.17.

An element in a graded ring is called decomposable if it can be written as a sumof nontrivial products of elements of nonzero degree.

18

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

Lemma 4.6 ([28, Proposition 6.3]). Let P be a flag 3-polytope and K its dualLe∗simplicial complex. Then the ring H ∗ (ZP ) ∼=J⊂[m] H (KJ ) is multiplicativelyLe 0 (KJ ).
generated by
HJ⊂[m]

To prove this lemma it is enough to show that each nontrivial cohomology classe 1 (KI ) ⊂ H ∗ (ZP ) is decomposable or, equivalently, the product map
in HM
e 0 (KI ) ⊗ H
e 0 (KI ) → H
e 1 (KI )H
1
2
I=I1 tI2

is surjective. This proof is quite technical. We include it in Appendix B for thereader’s convenience.
Lemma 4.7. A simple 3-polytope P 6= ∆3 with m facets is flag if and only if
any nontrivial cohomology class in H m−2 (ZP ) is decomposable. In particular, if
H m−2 (ZP ) = 0 then either P is flag or P = ∆3 .
Proof. Suppose that P is not flag. Since P 6= ∆3 , it has a 3-belt {Fj1 , Fj2 , Fj3 }.
Equivalently, the dual complex K has a missing 3-face J = {j1 , j2 , j3 }. It givesa nonzero cohomology class α ∈ H −1,2J (ZP ) ⊂ H 5 (ZP ). Consider the Poincaré
duality pairing
H m−2 (ZP ) ⊗ H 5 (ZP ) → H m+3 (ZP ) = Z,
which specifies toH −(m−4),2([m]\J) (ZP ) ⊗ H −1,2J (ZP ) → H −(m−3), 2[m] (ZP ) = Z(see Proposition 4.5). Take β ∈ H −(m−4),2([m]\J) (ZP ) ⊂ H m−2 (ZP ) such that α · β
is a generator of H −(m−3), 2[m] (ZP ) = Z. By Theorem 2.17, H −(m−4),2([m]\J) (ZP ) =e 0 (K[m]\J ), and any element of H
e 0 (K[m]\J ) is indecomposable by Proposition 4.5.
H
We have therefore found an indecomposable element β ∈ H m−2 (ZP ).
Now suppose that P is flag. By Proposition 4.5,MM
e 0 (KI ) ⊕e 1 (KI ),H m−2 (ZP ) =H
H|I|=m−3|I|=m−4e 0 (KI ) ⊗ H
e 1 (K[m]\I ) → Z. Since K is
Consider the Poincaré duality pairing H
1e (K[m]\I ) = 0 for |I| = m − 3 (as there are no missing 2-faces). Hence,
flag, H
L
L
m−2
e0
e1
(ZP ) =
|I|=m−3 H (KI )=0 by Poincaré duality, and H|I|=m−4 H (KI ).
m−2
Then each nonzero element of H
(ZP ) is decomposable by Lemma 4.6.
Theorem 4.8. Let P be a flag 3-polytope, and assume given a ring isomorphism
H ∗ (ZP ) ∼= H ∗ (ZP 0 ) for another simple 3-polytope P 0 . Then P 0 is also flag.
In other words, the property of being a flag 3-polytope is B-rigid.
Proof. This follows from Lemma 4.7 and the fact that ∆3 is B-rigid.

Remark. Theorem 4.8 also follows from [28, Theorem 6.6].
Proposition 4.9. Let P be a simple 3-polytope.
(a) The product H 3 (ZP ) ⊗ H 3 (ZP ) → H 6 (ZP ) is trivial if and only if P does
not have 4-belts.
(b) The triple Massey product H 3 (ZP ) ⊗ H 3 (ZP ) ⊗ H 3 (ZP ) → H 8 (ZP ) is
trivial if P does not have 4-belts.
Proof. We first prove (a). Suppose P has a 4-belt (F1 , F2 , F3 , F4 ). It corresponds toa chordless 4-cycle {1, 2, 3, 4} in K = KP , i. e. a cycle with {1, 3} ∈/ K and {2, 4} ∈/ K.
e 0 (K{1,3} )⊗ H
e 0 (K{2,4} ) → H
e 1 (K{1,2,3,4} ), andHence, we have a nontrivial product H
a nontrivial product H 3 (ZP ) ⊗ H 3 (ZP ) → H 6 (ZP ).

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES19

Now suppose there is a nontrivial product H 3 (ZP ) ⊗ H 3 (ZP ) → H 6 (ZP ). We
L
L
e0
e1
e0
have H 6 (ZP ) =|I|=5 H (KI ) ⊕|I|=4 H (KI ). Elements of H (KI ) are indee 1 (KI ) with |I| = 4 can be decomposed into a productcomposable. An element of H
if and only if I can be split into two pairs of non-adjacent vertices, which meansthat I is a chordless 4-cycle. It corresponds to a 4-belt in P .
To prove (b), assume that there is a nontrivial Massey product hα, β, γi ∈H 8 (ZP ). Then, by Theorem 2.20, the graph KP1 contains an induced subgraphisomorphic to one of the five graphs in Figure 1. By inspection, each of these fivegraphs has a chordless 4-cycle (the outer cycle for the first four graphs, and the left
cycle for the last one). Hence, the polytope P has a 4-belt.

It is not known whether moment-angle manifolds of polytopes from the Pogorelov
class P have nontrivial Massey products of cohomology classes of dimension > 3 orof order > 3, or whether these moment-angle manifolds are formal in the sense of
rational homotopy theory. For general polytopes P , there are examples of nontrivial
Massey products of any order in H ∗ (ZP ), see [34].
Theorem 4.10. Let P be a simple 3-polytope without 4-belts, and assume given aring isomorphism H ∗ (ZP ) ∼= H ∗ (ZP 0 ) for another simple 3-polytope P 0 . Then P 0
also does not have 4-belts.
It other words, the property of being a 3-polytope without 4-belts is B-rigid.
Proof. This follows from Proposition 4.9 (a).



Recently Fan, Ma and Wang proved that any polytope P ∈ P is B-rigid, see [27,Theorem 3.1]. The proof builds upon the following crucial lemma:
Lemma 4.11 ([27, Corollary 3.4]). Consider the set of cohomology classes
T (P ) = {±[ui vj ] ∈ H 3 (ZP ),Fi ∩ Fj = ∅}.
∼=If P ∈ P, then for any cohomology ring isomorphism ψ : H ∗ (ZP ) −→ H ∗ (ZP 0 ), we
have ψ(T (P )) = T (P 0 ).
Note that the lemma above does not hold for all simple 3-polytopes. For example,
if P is a 3-cube with the pairs of opposite facets {F1 , F4 }, {F2 , F5 }, {F3 , F6 }, then∼=ZP ∼= S 3 × S 3 × S 3 and there is an isomorphism ψ : H ∗ (ZP ) −→ H ∗ (ZP ) whichmaps [u1 v4 ] to [u1 v4 ] + [u2 v5 ].
We include the proof of Lemma 4.11 in Appendix C for the reader’s convenience,
and also because some details were missing in the original argument. Note that thisproof uses Theorem 4.8 and Theorem 4.10.
5. Main results
Here we prove the cohomological rigidity for small covers and quasitoric manifolds over 3-polytopes from the Pogorelov class P. We start with a crucial lemma.
Lemma 5.1. In the notation of Theorem 2.3, consider the set of cohomology classes
D(M ) = {±[vi ] ∈ H 2 (M ),i = 1, . . . , m}.

If P is a 3-polytope from the Pogorelov class P and M 0 is a quasitoric manifold∼=
over P 0 , then for any cohomology ring isomorphism ϕ : H ∗ (M ) −→ H ∗ (M 0 ) we0
have ϕ(D(M )) = D(M ). Moreover, the set D(M ) is mapped bijectively under ϕ.
Proof. The idea is to reduce the statement to Lemma 4.11. The ring isomorphism∼=
ϕ is determined uniquely
H 2 (M ) −→ H 2 (M 0 ) of free abelian
Pmby the isomorphism
0groups. Let ϕ([vi ]) = j=1 Aij [vj ] for some Aij ∈ Z, 1 6 i, j 6 m. The elements
Aij are not defined uniquely as there are linear relations between the classes [vj0 ] in

20

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARKH 2 (M 0 ). To get rid of this indeterminacy, one can choose a vertex x = Fi1 ∩Fi2 ∩Fi3
of P and a vertex x0 = Fp0 1 ∩ Fp0 2 ∩ Fp0 3 of P 0 . Then the complementary cohomology
classes [vi ] with i ∈/ {i1 , i2 , i3 } form a basis in H 2 (M ) and the cohomology classes[vp0 ] with p ∈/ {p1 , p2 , p3 } form a basis in H 2 (M 0 ), so we have
(5.1)

m
X

ϕ([vi ]) =Bip [vp0 ],i ∈ [m] \ {i1 , i2 , i3 },p∈{p/ 1 ,p2 ,p3 }with uniquely defined Bip ∈ Z, i ∈ [m] \ {i1 , i2 , i3 }, p ∈ [m] \ {p1 , p2 , p3 }.
As we have seen in the proof of Proposition 3.5, the isomorphism ϕ gives an∼=isomorphism ψ : H ∗ (ZP ) −→ H ∗ (ZP 0 ), which is obtained from (3.2) by passing to∼=
the cohomology. We write (3.2) as ϕe : C(P, Λ) −→ C(P 0 , Λ0 ). This isomorphism isdefined on the exterior generators ui and the polynomial generators vi of the Koszul
algebra C(P, Λ) by the same formulae as ϕ.
Now take a cohomology class [ui vj ] ∈ H 3 (ZP ). By Lemma 4.11, it is mapped
under ψ to an element ε[u0k vl0 ] ∈ H 3 (ZP 0 ), ε = ±1. Choose vertices x = Fi1 ∩Fi2 ∩Fi3
/ Fi ∪ Fj and x0 ∈/ Fk0 ∪ Fl0 (seeof P and x0 = Fp0 1 ∩ Fp0 2 ∩ Fp0 3 of P 0 such that x ∈0Lemma 4.3). We use the vertices x and x to choose bases in the groups H 2 (M )and H 2 (M 0 ) as described in the first paragraph of the proof. Then we haveXψ[ui vj ] =Bip Bjq [u0p vq0 ].
p,q ∈{p/ 1 ,p2 ,p3 }On the other hand, we have ψ[ui vj ] = ε[u0k vl0 ] by Lemma 4.11. Hence,Xa=
Bip Bjq u0p vq0 − εu0k vl0p,q ∈{p/ 1 ,p2 ,p3 }is a coboundary in C(P 0 , Λ0 ), so there exists
Xc=
Lpq u0p u0q ,dc = a.

p,q ∈{p/ 1 ,p2 ,p3 }, p<q

We havedc =X

Lpq (u0q vp0 − u0p vq0 ).

p,q ∈{p/ 1 ,p2 ,p3 }, p<qComparing this with the expression for a we obtain the following relations between
the coefficients:
Bip Bjq = −Biq Bjp = −Lpq for p < q and {p, q} =6 {k, l};(−Lkl if k < l,
(5.2)
Bik Bjl − ε = −Bil Bjk =Llk
if l < k;Bip Bjp = 0.
From the third equation of (5.2) we have, for any p ∈ [m]\{p1 , p2 , p3 }, either Bip = 0
orThe first
 Bjp = 0.  equation of (5.2) implies that for {p, q} 6= {k, l} the vectors
Bip
Biq
and
are linearly dependent. Hence, for {p, q} 6= {k, l}, either one
Bjp
−Bjq




Bip
Biq
of the vectors
andis zero, or both vectors are nonzero and have a
Bjp
Bjq
zero entry on the same place.
From the second
equation Bik Bjl + Bil Bjk = ε we see


 
Bik
Bil
that both vectors bk =and bl =are nonzero. If there is a nonzero
Bjk
Bjl


Bip
vector bp =for some p ∈/ {k, l}, then by considering the pairs (bp , bk ) andBjp

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

21(bp , bl ) we see that both bk and bl have zero on the same place, which contradicts
the second equation of (5.2). It follows that Bip = Bjp = 0 for any p ∈/ {k, l}, and




 
 
Bik
0
0 Bil
Bik Bil
Bik Bil
or
.
=
=
ε
ε
0
0
Bjk Bjl
Bjk Bjl
Bik
Bil
Since all entries are integer, we have Bik = ±1 and Bil = ±1. Then (5.1) gives
ϕ([vi ]) ∈ {±[vk0 ], ±[vl0 ]}. It follows that ϕ(D(M )) ⊂ D(M 0 ).
It remains to show that the set D(M ) is mapped bijectively under ϕ. For this
we note that [vi ] 6= ±[vj ] in H 2 (M ) for any i 6= j. Indeed, by Lemma 4.3 we canchoose a vertex x ∈/ Fi ∪ Fj . Then both [vi ] and [vj ] belong to a basis of H 2 (M ).
Now, since ϕ is an isomorphism, we also have ϕ([vi ]) 6= ±ϕ([vj ]) in H 2 (M 0 ). Thus,
both sets D(M ) and D(M 0 ) consist of 2m elements and ϕ(D(M )) = D(M 0 ).
It follows from the Steinitz Theorem that any toric manifold of complex dimension 3 is a quasitoric manifold. Also, the family of quasitoric manifolds agrees withthat of topological toric manifolds in real dimension 6 if we forget the actions.
Now we state the first main result.
Theorem 5.2. Let M = M (P, Λ) and M 0 = M (P 0 , Λ0 ) be quasitoric manifolds
over 3-dimensional simple polytopes P and P 0 , respectively. Assume that P belongs
to the Pogorelov class P. Then the following conditions are equivalent:∼=(a) there is a cohomology ring isomorphism ϕ : H ∗ (M ) −→ H ∗ (M 0 );(b) there is a diffeomorphism M ∼= M 0;(c) there is an equivalence of characteristic pairs (P, Λ) ∼ (P 0 , Λ0 ).
Proof. The implication (b)⇒(a) is obvious. The implication (c)⇒(b) follows from
Proposition 2.21. We need to prove (a)⇒(c).
0
By Lemma 5.1, ϕ([vi ]) = ±[vσ(i)], where σ is a permutation of the set [m].
Renumbering the facets and multiplying the matrix Λ from the right by a matrix
B as in Definition 2.6, we may assume that ϕ([vi ]) = vi0 ; this does not change
the equivalence class of the pair (P, Λ). Then ϕ[vi vj ] = [vi0 vj0 ]. By Corollary 2.4,
[vi vj ] = 0 in H ∗ (M ) if and only if Fi ∩ Fj = ∅ and [vi vj vk ] = 0 in H ∗ (M ) if andonly if Fi ∩ Fj ∩ Fk = ∅ in P , and the same holds for H ∗ (M 0 ) and P 0 . It follows
that KP is isomorphic to KP 0 . Hence, P and P 0 are combinatorially equivalent.
Now consider the (3 × m)-matrices Λ and Λ0 . First, by changing the order offacets in P and P 0 if necessary we may assume that F1 ∩ F2 ∩ F3 6= ∅ in P andF10 ∩ F20 ∩ F30 6= ∅ in P 0 . Then, by multiplying the matrices Λ and Λ0 from the left
by appropriate matrices from GL (3, Z) we may assume that

1 0 0 λ14 · · · λ1m
1 0 0 λ014 · · · λ01m
Λ = 0 1 0 λ24 · · · λ2m  ,Λ0 = 0 1 0 λ024 · · · λ02m  .
0 0 1 λ34 · · · λ3m
0 0 1 λ034 · · · λ03m
This does not change the equivalence class of pairs (P, Λ) and (P 0 , Λ0 ). Now theentries λjk , 4 6 k 6 m, are the coefficients in the expression of [vj ], 1 6 j 6 3, via
the basis [v4 ], . . . , [vm ] of H 2 (M ). The same holds for the λ0jk . Since ϕ([vi ]) = vi0 , itfollows that λjk = λ0jk . Thus, the pairs (P, Λ) and (P 0 , Λ0 ) are equivalent.

Remark. Any smooth structure on a quasitoric manifold M over a polytope P ∈ P isequivalent to the standard one defined on the canonical model M (P, Λ) via Proposition 2.21. This follows from the general classification results for 6-dimensional
manifolds, see Corollary 6.4.
Corollary 5.3. Toric, quasitoric and topological toric manifolds over polytopes
from the Pogorelov class P are cohomologically rigid.

22

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

The family of quasitoric (or topological toric) manifolds over 3-polytopes fromthe Pogorelov class P is large enough, as there is at least one quasitoric manifold
over any such polytope by Proposition 2.8 (recall that it uses the Four Colour Theorem). There are fewer toric manifolds in this family. In fact, there are no projective
toric manifolds over combinatorial flag 3-polytopes without 4-belts. The reason isthat a Delzant 3-polytope must have at least one triangular or quadrangular faceby the result of C. Delaunay [23] (see also [2]). On the other hand, there are many
non-projective toric manifolds in this family, see [42].
Remark. In general, a (non-equivariant) diffeomorphism M ∼= M 0 does not imply an equivalence of characteristic pairs. For example consider the family of toric
manifolds (Hirzebruch surfaces) Hk = CP (O(k) ⊕ C), where O(k) is the kth powerof the canonical line bundle over CP 1 , C is a trivial line bundle, and CP (−) denotes the complex projectivisation. Then each Hk is a quasitoric manifold over a
combinatorial 4-gon with characteristic matrix
1 0 −1 k.
0 1 0 −1
Manifolds Hk with even k are all diffeomorphic to S 2 × S 2 , but the characteristic
matrices corresponding to different positive k are not equivalent. Similar examples
exist in all dimensions, see e.g. [36].
Our second main result is about small covers (or hyperbolic 3-manifolds).
Theorem 5.4. Let N = N (P, Λ) and N 0 = N (P 0 , Λ0 ) be small covers of 3dimensional simple polytopes P and P 0 , respectively. Assume that P belongs to
the Pogorelov class P, so N is a hyperbolic 3-manifold of Löbell type. Then thefollowing conditions are equivalent:∼=(a) there is a cohomology ring isomorphism ϕ : H ∗ (N ; Z2 ) −→ H ∗ (N 0 ; Z2 );(b) there is a diffeomorphism N ∼= N 0;(c) there is an equivalence of Z2 -characteristic pairs (P, Λ) ∼ (P 0 , Λ0 ).
Proof. We only need to prove the implication (a)⇒(c). Using Proposition 2.10 weupgrade (P, Λ) and (P 0 , Λ0 ) to Z-characteristic pairs and consider the corresponding
quasitoric manifolds M = M (P, Λ) and M 0 = M (P 0 , Λ0 ). Since the cohomology ringH ∗ (M ; Z2 ) is obtained from H ∗ (N ; Z2 ) by doubling the grading (see Theorem 2.9),∼=
we have an isomorphism H ∗ (M ; Z2 ) −→ H ∗ (M 0 ; Z2 ). Now the equivalence of characteristic pairs follows from Theorem 5.2 (with coefficients in Z2 ).

6. Classification of 6-dimensional manifolds and related problems
The classification of smooth simply connected 6-dimensional manifolds withtorsion-free homology was done in the works of Wall [44] and Jupp [32]. They
also stated a result in the topological category, whose proof was corrected in thework of Zhubr [45]. The latter work also treated the case of homology with torsion. We only give the following result, which will be enough for our purposes (thecohomology is with integer coefficients, unless otherwise specified).
∼=

Theorem 6.1 ([44], [32]). Let ϕ : H ∗ (N ) −→ H ∗ (N 0 ) be an isomorphism of thecohomology rings of smooth closed simply connected 6-dimensional manifolds N, N 0
with H 3 (N ) = H 3 (N 0 ) = 0. Suppose that(a) ϕ(w2 (N )) = w2 (N 0 ), where w2 (N ) ∈ H 2 (N ; Z2 ) is the second Stiefel–
Whitney class;(b) ϕ(p1 (N )) = p1 (N 0 ), where p1 (N ) ∈ H 4 (N ) is the first Pontryagin class.
Then the manifolds N and N 0 are diffeomorphic.

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES23

The following lemma is proved using Steenrod squares:
Lemma 6.2 ([17, Lemma 8.1]). Suppose that the ring H ∗ (N ; Z2 ) is generated by H k (N ; Z2 ) for some k > 0. Then any cohomology ring isomorphism∼=
ϕ : H ∗ (N ; Z2 ) −→ H ∗ (N 0 ; Z2 ) preserves the total Stiefel–Whitney class, i. e.
0ϕ(w(N )) = w(N ).
Lemma 6.2 applies to toric or quasitoric manifolds, whose cohomology is generated in degree two. From Theorem 6.1 we obtain∼=

Corollary 6.3. Let ϕ : H 2 (M ) −→ H 2 (M 0 ) be an isomorphism of second cohomology groups of 6-dimensional smooth quasitoric manifolds. Suppose that(a) ϕ preserves the cubic form H 2 (M )⊗H 2 (M )⊗H 2 (M ) → Z = H 6 (M ) givenby the cohomology multiplication;(b) ϕ preserves the first Pontryagin class.
Then the manifolds M and M 0 are diffeomorphic.
From the topological invariance of rational Pontryagin classes (proved in general
by S. P. Novikov) we obtain
Corollary 6.4. Let M and M 0 be 6-dimensional smooth quasitoric manifolds. If
M and M 0 are homeomorphic, then they are diffeomorphic.
The characteristic classes of quasitoric manifolds are given as follows:
Proposition 6.5 ([22, Corollary 6.7]). In the notation of Theorem 2.3, the total
Stiefel–Whitney and Pontryagin classes of a quasitoric manifold M are given bymm
YY(1 + vi2 ).
(1 + vi ) mod 2,p(M ) =
w(M ) =i=1

i=1

2
In particular, w2 (M ) = v1 + · · · + vm mod 2, and p1 (M ) = v12 + · · · + vm.

Corollary 6.6. A family of 6-dimensional quasitoric manifolds is cohomologicallyrigid if any cohomology ring isomorphism between manifolds from the family preserves the first Pontryagin class.
This reduces cohomological rigidity for 6-dimensional quasitoric manifolds M toa problem of combinatorics and linear algebra, as both the cohomology ring H ∗ (M )2
are defined entirely in terms
and the first Pontryagin class p1 (M ) = v12 + · · · + vmof the characteristic pair (P, Λ).
Our result on cohomological rigidity for quasitoric manifolds over polytopes fromthe Pogorelov class P (Theorem 5.2) gives a complete classification for this particular class of simply connected 6-manifolds, and its proof is indepenent of the generalclassification results of Wall and Jupp. The invariance of the first Pontryagin classfor quasitoric manifolds over polytopes from the Pogorelov class follows directlyfrom Lemma 5.1. It would be interesting to find a direct (combinatorial?) proof ofthis fact. Bott towers (of any dimension) form another family of toric manifolds forwhich the invariance of Pontryagin classes is known, see [15].
Remark. In dimension 4 we have the identity hp1 (M ), [M ]i = 3 sign(M ), where[M ] ∈ H4 (M ) is the fundamental class and sign(M ) is the signature of M . Therefore, p1 is invariant under cohomology ring isomorphisms. When M is a toric manifold, the signature is equal to 4−m, where m is the number of vertices in the quotientpolygon P (see e.g. [11, Example 9.5.3]). The identity hp1 (M ), [M ]i = 3 sign(M )then becomes2
hv12 + · · · + vm
, [M ]i = 12 − 3m,
which can be seen directly from Theorem 2.2.

24

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

Appendix A. Belts in 3-polytopes
Here we give proofs of two combinatorial lemmata on belts in flag 3-polytopes,
originally due to [28] and [27] respectively. These proofs are included mainly for thesake of completeness, but we also fill in some details missing in the original works.
Lemma A.1 is used in the proof of the product decomposition lemma in Appendix B,
while Lemma A.3 is used in the proof of B-rigidity of the set of canonical generatorsof H 3 (ZP ) in Appendix C.
Recall that a belt in a simple polytope P corresponds to a chordless cycle inthe dual simplicial complex KP , or to a full subcomplex (KP )I isomorphic to theboundary of a polygon.
Lemma A.1. Let P be a flag simple 3-polytope. Then for every three facets Fi ,Fi0 , Fk with Fi ∩ Fi0 = ∅, there exists a belt B such that Fi , Fi0 ∈ B and Fk ∈/ B.
We reformulate this lemma in the dual notation; this is how the lemma wasstated and proved in [28]:Lemma A.2 ([28, Lemma 6.1]). Let K be a flag triangulation of the disk D2
with m vertices, and let S be the set of vertices of the boundary ∂K. Assume that
KS = ∂K. Then for every missing face {i, i0 } of K there exists a subset I ⊂ [m]such that {i, i0 } ⊂ I and KI is a chordless cycle (boundary of a polygon).
To obtain Lemma A.1 from Lemma A.2 we take K = (KP )[m]\{k} , the simplicial
complement to the vertex of KP corresponding to the facet Fk ⊂ P . Then K isa flag triangulation of D2 (as a full subcomplex in the flag complex KP ), andKS = ∂K because KP is flag. Lemma A.2 gives a chordless cycle I in K ⊂ KP ,which corresponds to the required belt in P .
The star and link of a vertex {i} ∈ K are the subcomplexes
starK {i} = {I ∈ K : {i} ∪ I ∈ K},

linkK {i} = {I ∈ K : {i} ∪ I ∈ K, i ∈/ I}.

Proof of Lemma A.2. We use the induction on m, the number of vertices of K.
Since K is flag, ∂K has at least 4 vertices, that is, |S| > 4 and m > 5. If m = 5,then |S| = 4 and K is the cone over a square. Then we can take I = S, as every
missing face of K is contained in the chordless boundary cycle KS = ∂K.
Now assume that the statement holds for simplicial complexes with < m vertices.
If both vertices i and i0 lie in ∂K, then I = S is the required chordless cycle. Hence,
we only need to consider the case {i, i0 } 6⊂ S. If |S| > m−1, then there is no missing
face {i, i0 } such that {i, i0 } 6⊂ S. Hence, |S| < m − 1. For a vertex j ∈ S, denote by
mj the number of vertices in starK {j}. Then mj > 4 for any j ∈ S, since KS = ∂K.
I. Suppose that there is a vertex j ∈ S \ {i, i0 } such that mj = 4. Then the setof vertices of starK {j} is {j, j 0 , j 00 , k}, where j, j 0 , j 00 ∈ S and k ∈/ S, see Figure 2.
j

j0

j

j 00

k

j0

j 00

k

Figure 2. starK {j} and its bistellar 1-move(i) If there is no vertex k 0 ∈ S \ {j, j 0 , j 00 } such that {k, k 0 } is an edge of K, thenthe simplicial complex K0 = K[m]\{j} satisfies the hypothesis of the lemma. Hence,
there is a subset I 0 of [m] \ {j} such that {i, i0 } ⊂ I 0 and KI0 0 is a chordless cycle.
Then I = I 0 is the required set, as KI 0 = KI0 0 .

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

25(ii) Now assume that there exists a vertex k 0 ∈ S \ {j, j 0 , j 00 } such that {k, k 0 }is an edge in K. Let K0 be the simplicial complex obtained from K by applying
0
a bistellar 1-move at starK {j}, see Figure 2. Then K00 := K[m]\{j}satisfies the00
hypothesis of the lemma. By induction, there is a subset I of [m] \ {j} such that{i, i0 } ⊂ I 00 and KI0000 is a chordless cycle. If j 0 or j 00 is not in I 00 , then I = I 00 is therequired set. If both j 0 and j 00 are in I 00 , then I = I 00 ∪ {j} is the required set.
II. Suppose that mj > 4 for every j ∈ S \ {i, i0 }. Let S = {j1 , . . . , jn }, orderedcounterclockwise, and assume that j1 ∈/ {i, i0 }. Let Vjp denote the set of vertices of
starK (jp ), so |Vjp | = mjp , for 1 6 p 6 n. Note that if jp ∈ S \ {i, i0 }, then mjp > 4
and |Vjp \ S| > 1.
(i) Assume that, for some jp ∈ S \ {i, i0 }, there is no edge {k, k 0 } in K such that(∗)

k ∈ Vjp \ S

and k 0 ∈ S \ {jp−1 , jp , jp+1 },where j0 = jn .

Then K0 := K[m]\{jp } satisfies the hypothesis of the lemma, so we can find therequired subset I of [m] \ {jp }.
(ii) Assume that, for every jp ∈ S \ {i, i0 }, there is an edge {kp , jqp } in K satisfying (∗) for k = kp and k 0 = jqp . We shall lead this case to a contradiction.
Set I1 = {j1 , k1 , jq1 }. Then KI1 divides K into two simplicial complexes K1 andK2 , where K1 has boundary vertices j1 , . . . , jq1 , k1 , and K2 has boundary vertices
jq1 , . . . , jn , k1 , see Figure 3.
j1

j2

jn

k1
jq1 −1

jq1 +1

jq1

Figure 3. KI1 divides K into two simplicial complexes.
Since {i, i0 } 6⊂ S, either {i, i0 } ∩ {j1 , . . . , jq1 −1 } = ∅ or {i, i0 } ∩ {jq1 , . . . , jn } = ∅.
Without loss of generality, assume that {i, i0 } ∩ {j1 , . . . , jq1 −1 } = ∅. Then mjp > 4
for 1 6 p 6 q1 − 1. By the flagness of K and the condition for the existence of anedge satisfying (∗), there is no vertex k ∈ [m] \ S such that k is connected to thevertices jp and jp+2 for 1 6 p 6 q1 − 2. This implies in particular that q1 > 3.
Now consider the path from j2 to k2 and to jq2 . If k2 = k1 , then we may assume
that jq2 = jq1 . Otherwise, k2 must be contained in the simplicial complex K1 . In
either case, the the path j2 − k2 − jq2 is contained in the simplicial subcomplex K1
with boundary vertices j1 , . . . , jq1 , k1 . Proceeding inductively, we obtain that thepath jp −kp −jqp is contained in the simplicial subcomplex whose boundary vertices
are jp−1 , . . . , jqp−1 , kp−1 , see Figure 4. It follows that p < qp 6 qp−1 6 · · · 6 q1 .
Eventually we obtain p such that qp = p + 2, so the vertex kp is connected to thevertices jp and jp+2 . This is a contradiction.
From I and II, the lemma is proved.
Lemma A.3 ([27, Lemma 3.2]). Let P be a flag 3-polytope without 4-belts. Thenfor every three different facets Fi , Fi0 , Fk with Fi ∩ Fi0 = ∅ there is a belt B suchthat Fi , Fi0 ∈ B, Fk ∈/ B, and Fk does not intersect at least one of the two connectedcomponents of B \ {Fi , Fi0 }.

26

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK
jp−1
jp
kp−1

jqp−1

Figure 4. The path jp − kp − jqp is contained in the above simplicial complex.
Proof. We work with the dual simplicial complex K = KP , which is a triangulated
2-sphere. We need to find a subset I ⊂ [m] \ {k} such that {i, i0 } ⊂ I, KI is ae 0 (K(I\{i,i0 })∪{k} ) 6= 0. By Lemma A.2, there is a subset I0
chordless cycle, and Hof [m] \ {k} such that {i, i0 } ⊂ I0 and KI0 is a chordless cycle. We construct therequired subset I by modifying I0 .
i

i

KI0

Kin
e
X
i0

i

Kout
Ye

k

X

i0

Y

k

i0

e Ye , X and Y
Figure 5. Examples of KI0 , Kin , Kout , X,
Since KI0 is a cycle, it divides K into two polygons (triangulated discs) Kin andKout with the common boundary KI0 . Assume that the vertex k is contained in Kin .
e andThe vertices i and i0 divide the cycle KI0 into two arcs, and we denote by X
e t Ye .
Ye the sets of vertices in I0 \ {i, i0 } contained in these arcs, so I0 \ {i, i0 } = Xee
We set X = linkK {k} ∩ X and Y = linkK {k} ∩ Y , see Figure 5. If either X or Y
e 0 (K(I \{i,i0 })∪{k} ) 6= 0, so I = I0 is the required subset. In whatis empty, then H
0follows we assume that both X and Y are nonempty.
We consider the links of all x ∈ X in Kout . Since KI0 is a chordless cycle,
every such link has at least three vertices, that is, there is a vertex in linkKout {x}which is not in I0 . To simplify notation, for X ⊂ [m], we write linkK X instead ofSx∈X linkK {x}. Now definee ∪ {i, i0 } ∪ linkK X.
KX := the full subcomplex of K induced on the set X
out
We take the outermost path PX between i and i0 in KX with respect to the vertex k,so that all vertices of KX not in PX are on the side of k, see Figure 6. Let IX be
the vertex set of PX .
i

i

i

k

k

i0

linkKout X

i0

KX

i0

The gray thick path is PX .

Figure 6. Examples of KX and PXCOHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

27

Claim. The full subcomplex of K induced on IX is the path PX , i.e., KIX = PX .
Proof of Claim. Suppose to the contrary that there is a subset {p, q, r} of I1
e Note thatsuch that K{p,q,r} is a triangle. Consider the intersection {p, q, r} ∩ X.
e|{p, q, r} ∩ X| < 3 because KXe is a part of a chordless cycle KI0 . We have thefollowing cases, shown in Figure 7.
q

r

r

r

r
r

q
x
p

x

q
x

0

p

q

q

x

(1)

p

(2)-(a)

x00

p

x0

p

x

x

x0

(2)-(b)(3)-(a)(3)-(b)Figure 7. KX has no triangle.
e = 2, say {p, q, r} ∩ X
e = {p, q}, then p and q are consecutive(1) If |{p, q, r} ∩ X|
vertices in X, and r is in linkKout {x} for some x ∈ X. Then, p or q is on
the side of k in KX . This is a contradiction.
e = 1, say {p, q, r} ∩ X
e = {p}, then q ∈ linkK x and r ∈(2) If |{p, q, r} ∩ X|
200
linkK2 {x } for some x, x ∈ X.
(a) If x 6= x0 , then p must be on the side of k in KX , which contradicts
the assumption that p ∈ PX .
(b) If x = x0 , then either q or r is on the side of k in KX , and we obtaina contradiction again.
e = 0, then there are x, x0 , x00 in X such that p ∈ linkK {x},(3) If |{p, q, r} ∩ X|outq ∈ linkKout {x0 }, and r ∈ linkKout {x00 }. Since p, q, r are in the outermost
path PX , the case x = x0 = x00 is impossible. Hence, we may assume thatx 6= x0 or x 6= x00 .
(a) If x, x0 , x00 are all distinct, then one of p, q, and r must be on the sideof k in KX , which contradicts the assumption that p, q, r are on PX
and PX is the outermost path with respect to k.
(b) If x0 = x00 , then either q or r is on the side of k in KX . This final
contradiction finishes the proof of the claim.
We return to the proof of Lemma A.3. The endpoints of the path PX = KIX arei, i0 and there is no edge connecting k and IX . Therefore, if KIX ∪Ye is a chordlesscycle, then IX ∪ Ye is the required set I.
Suppose that KIX ∪Ye has a chord. Then the chord must be an edge in Kout .
Note that since K has no chordless 4-cycles, there is no edge connecting linkKout X
and Y . We consider the vertices x+ ∈ X and x− ∈ X that are closest to i and i0 ,e Similarly, consider the vertices y+ ∈ Y andrespectively, on the arc containing X.
0y− ∈ Y that are closest to i and i , respectively, on the arc containing Ye . Denote bye lying strictly between i and x+ . Define the subsets
X+ the subset of vertices in Xeee
X− ⊂ X, Y+ ⊂ Y and Y− ⊂ Y similarly. See Figure 8, left.
We consider two cases.
Case 1. There is no edge connecting IX and Y− ∪ Y+ in Kout .
We define
KY := the full subcomplex of K induced on Ye ∪ {i, i0 } ∪ linkK (Ye \ (Y ∪ Y− ∪ Y+ )).
in28

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK
i

i

i

x+
y+

y+

y+

k

k

x−

y−

i0

i0

y−

i0

KY

linkKin (Ye \ (Y ∪ Y− ∪ Y+ ))y−

The gray thick path is PY .

Figure 8. Example of Case 1
We take the innermost path PY connecting i and i0 in KY with respect to k, see
Figure 8, and let IY be the vertex set of PY . Then KIY = PY by the same argument
as the claim above, and IX ∪ IY is the required subset I.
i

i

i
x+

y+
x+

y+
k

y−

x−

i0
x−

y−

i0

KY+ and PY+

i0

+
+
KX
and PX

Figure 9. Example of Case 2
Case 2. There is an edge connecting IX and Y+ or Y− in Kout .
Suppose that IX is connected by an edge in Kout to only one of Y+ and Y− , say
to Y+ . We define
K+ = the full subcomplex of K induced on Ye ∪ {i, i0 } ∪ linkK (Ye \ (Y ∪ Y− )).
Yin

We take the innermost path PY+ connecting i and i0 in Kin with respect to thevertex k, and let IY+ be the vertex set of PY+ . See Figure 9, middle. Then KI + = PY+
Y
by the same reason as the claim above. If KIX ∪I + is a chordless cycle, then IX ∪ IY+
Y
is the required subset I.
If KIX ∪I + has a chord, then it must be an edge in Kin connecting linkKin Y+ andY
X+ ∩ IX . In this case we modify IX as follows. We definee ∪ {i, i0 } ∪ linkK (X ∪ X+ ).
K+ := the full subcomplex of K induced on X
X

out

+
We take the outermost path PX+
connecting i and i in KX
with respect to the+++vertex k, see Figure 9, right. Let IX be the vertex set of PX . Then KI + = PXbyX+
the same argument as the claim above, and we can see that IX ∪ IY is the required
0

subset I. Indeed, we only need to check that there is no edge connecting linkKout X+
and Y in Kout . This is because there is an edge connecting IX and Y+ .
It remains to consider the case when IX is connected to both Y+ and Y− byedges in Kout . Here the same argument as above works if we consider
K± := the full subcomplex of K induced on Ye ∪ {i, i0 } ∪ linkK (Ye \ Y )
Y

in

and
±
e ∪ {i, i0 } ∪ linkK (X ∪ X+ ∪ X− )KX:= the full subcomplex of K induced on X
out+instead of KY+ and KX, respectively.

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

29

Appendix B. Proof of Lemma 4.6.
Here we give a proof which is different from the original proof of [28]. It uses areformulation of the description of product in the cohomology of a moment-angle
complex (Theorem 2.17) in terms of the polytope P . A detailed description of thisapproach can be found in [9, §5.8].
We need to prove that the product map
M
e 0 (KI ) ⊗ H
e 0 (KI ) → H
e 1 (KI )
(B.1)
H
1
2
I=I1 tI2

is surjective for any flag 3-polytope P and I ⊂ [m]. We first restate this in terms of
the polytope P rather than its dual simplical complex K. The decomposition of ∂P
into facets F1 , . . . , Fm defines a cell decomposition of ∂P which is Poincaré dual to
the simplicial decomposition K. The two decompositions have the same barycentric
subdivision, (∂P )0 ∼= K0 . We identify the set of facets {F1 , . . . , Fm } with [m], andfor each I ⊂ [m] define[PI =Fi ⊂ ∂P.
i∈I

Note that PI is the combinatorial neighbourhood of (KI )0 in K0 , so there is a'deformation retraction PI −→ KI . We have Poincaré duality isomorphisms(B.2)
H2−i (PI , ∂PI ) ∼= H i (KI ), i = 0, 1, 2,
where the boundary ∂PI consists of points x ∈ PI such that x ∈ Fj for some j ∈/ I.
Topologically, PI is a disjoint union of several discs with holes, and ∂PI is a disjoint
union of edge cycles.
The cellular homology groups Hi (PI , ∂PI ) have the following description. Let
PI = PI 1 t · · · t PI s be the decomposition into connected components. Then(a) H
P2 (PI , ∂PI ) is a free abelian group with basis of homology classes [PI k ] =i∈I k [Fi ], k = 1, . . . , s;sL(b) H1 (PI , ∂PI ) =H1 (PI k , ∂PI k ), where H1 (PI k , ∂PI k ) is a free abeliank=1group of rank one less the number of cycles in ∂PI k . A basis of H1 (PI k , ∂PI k )is given by any set of edge paths in PI k connecting one fixed boundary cyclewith the other boundary cycles.
As the product map (B.1) is stated in terms of the reduced cohomology groups
e i (KI ), we introduce the corresponding “reduced” homology groupsH(
i = 0, 1;b i (PI , ∂PI ) = Hi (PI , ∂PI ), P
H
H2 (PI , ∂PI )
[F
]
,
i
= 2.
i∈I i
Then we can rewrite (B.2) as(B.3)

b 2−i (PI , ∂PI ) ∼e i (KI ),H=H

i = 0, 1, 2.

With this interpretation in mind, we can rewrite the product map (B.1) as the“intersection pairing”
M
b 2 (PI , ∂PI ) ⊗ H
b 2 (PI , ∂PI ) → H
b 1 (PI , ∂PI ),
H
1
1
2
2
I=I1 tI2(B.4)[PI1p ] ⊗ [PI2q ] 7→ [PI1p ∩ PI2q ] = [γ1 ] + · · · + [γr ],where PI1p is a connected component of PI1 , PI2q is a connected component of PI2 ,and γ1 , . . . , γr are edge paths in P which form the connected components of theintersection PI1p ∩PI2q . (There is a sign involved in the transition from (B.1) to (B.4),
but it does not affect our subsequent considerations.)

30

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

Proof of Lemma 4.6. To see that (B.4) is surjective for a flag 3-polytope P , we recallsb 1 (PI , ∂PI ) = L H
b 1 (PI k , ∂PI k ) and consider for each connected component
that Hk=1

PI k of PI the decomposition ∂PI k = η1 t · · · t ηtk into boundary cycles. We mayb 1 (PI k , ∂PI k ) = 0. For each pairassume that tk > 2, as otherwise PI k is a disc and Hof boundary cycles ηp and ηq among η1 , . . . , ηtk , we shall decompose the generatorb 1 (PI k , ∂PI k ) corresponding to an edge path from ηp to ηq into a productgpq of H
b 2 (PI , ∂PI ) and Hb 2 (PI , ∂PI ), I1 t I2 = I. This will prove theof elements of H
1122
surjectivity of (B.4).
We choose facets Fp and Fq in ∂P \ PI k adjacent to ηp and ηq respectively, see
Figure 10. By Lemma A.1, there is a belt B = (Fj1 , . . . , Fjl ) with Fj1 = Fp and
p

PI

k

a
γ3

B1

a

γ2

Fp

q

B1

Fq

b
r

γ1

a
p

Figure 10. A belt crossing a disc with holes.
Fjr = Fq , where 3 6 r 6 l − 1. Let B1 = (Fj1 , . . . , Fjr ) be a part of the belt
between Fp and Fq (there are two such parts, and we can take any of them). The
complement ∂P \ B is a union of two open discs. We denote the closures of thesediscs by W1 and W2 ; each of them is a union of facets not in B. Now set
I1 = {i : Fi ∈ PI k ∩ B},X
b 2 (PI , ∂PI ),a=[Fi ] ∈ H
1
1
Fi ∈PI k ∩B1

I2 = I \ I1 ,Xb=b 2 (PI , ∂PI ).
[Fj ] ∈ H
2
2

Fj ∈PI k ∩W1Then a · b = [γ1 ] + · · · + [γs ] ∈ H1 (PI , ∂PI ), where each γi is an edge path in PI kthat begins at some boundary cycle ηji−1 and ends at ηji . We may assume that γ1
begins at ηp and γs ends at ηq (where ηp , ηq is the pair of boundary cycles chosenb 1 (PI , ∂PI ) is then equal to theabove). The homology class [γ1 ] + · · · + [γs ] ∈ Hbchosen generator gpq of H1 (PI k , ∂PI k ) corresponding to an edge path from ηp to
ηq . We have therefore decomposed gpq into a product a · b, as needed.

Appendix C. Proof of Lemma 4.11
The proof uses the combinatorial result of Lemma A.3 and an algebraic “annihilator lemma” of Fan, Ma and Wang.

COHOMOLOGICAL RIGIDITY OF TORIC MANIFOLDS OVER 3-POLYTOPES

31

Recall that the annihilator of an element r in a ring R is defined as
AnnR (r) = {s ∈ R : rs = 0}.
Lemma C.1 ([27, Lemma 3.3]). Let P be a 3-polytope from the Pogorelov class P,
with the dual complex K = KP . Let R = H ∗ (ZP ; k), where k is a field. In thenotation of Lemma 4.11, consider a k-linear combination of elements of T (P ),Xα=rij [ui vj ]{i,j}∈K/

with at least two nonzero rij ∈ k. Then, for any {k, l} such that rkl 6= 0,
dim AnnR [uk vl ] > dim AnnR α.
Proof. In view of the isomorphisms (B.3), we can rewrite the isomorphism of Theorem 2.17 asMb ∗ (PI , ∂PI )R = H ∗ (ZP ) ∼
H
=
I⊂[m]

(we omit the coefficient field k in the notation for homology).
Take a complementary subspace Lkl to AnnR [uk vl ] in R, so that Lkl ⊕AnnR [uk vl ] = R. For any β ∈ Lkl \ {0} we have β · [uk vl ] 6= 0. Furthermore, we canchoose Lkl respecting the multigrading, so that the Ith multigraded componentb ∗ (PI , ∂PI ) in Hb ∗ (PI , ∂PI ).
of Lkl is a complementaryPsubspace to AnnR [uk vl ] ∩ HThen we can write β =
β
,
where
β
denotestheIthmultigradedII⊂[m]\{k,l} I
component of β ∈ Lkl \ {0}. (Note that βI = 0 whenever I ∩ {k, l} =6 ∅, as such βI
would annihilate [uPk vl ].) We can choose I ⊂ [m] \ {k, l} such that βI · [uk vl ] 6= 0.
Now consider α =rij [ui vj ]. We claim that the (I ∪ {k, l})th multigraded component of β · α consists of βI · [uk vl ] only. Indeed, for any other component βI 0 of
β with I 0 6= I and any summand rij [ui vj ] of α, we have I 0 ∪ {i, j} 6= I ∪ {k, l}, asI 0 ∈ [m] \ {k, l}. Then (β · α)I∪{k,l} = βI · [uk vl ] 6= 0. Hence, Lkl ∩ AnnR α = {0},which implies that dim AnnR [uk vl ] > dim AnnR α.
In order to show that the strict inequality holds, we shall find an element ξ ∈AnnR [uk vl ] such that (Lkl ⊕ hξi) ∩ AnnR α = {0}. Take a summand rst [us vt ] of αdifferent from rkl [uk vl ]. That is, {s, t} =6 {k, l} and rst 6= 0. We can assume without
loss of generality that l ∈/ {s, t}. By Lemma A.3, there is a belt B in P such that
Fs , Ft ∈ B, Fl ∈/ B, and Fl does not intersect one of the two connected components
B1 and B2 of B \ {Fs , Ft }, say B1 . In the dual language, there is a chordless cycle
C in KP such that s, t ∈ C, l ∈/ C, and the vertex l is not joined by an edge to any
vertex of the connected component L1 of C \ {s, t}.
Now we observe that C \ {s, t} is a full subcomplex of KP and take ξ to be thee 0 (C \ {s, t}) ∼cohomology class in R = H ∗ (ZP ) given byP
a generator of H
= Z. Such agenerator is represented by the 0-cocycle i∈L1 α{i} (see Example 2.19). We haveP
ξ · [uk vl ] = 0 because we can write ξ = i∈L1 ±[uJi vi ] (see Example 2.19) andvi vl = 0 for any i ∈ L1 by the choice of the cycle C. On the other hand, the producte 1 (C) ∼ξ · [us vt ] corresponds to a generator of H
= Z. Therefore, ξ ∈ AnnR [uk vl ] andξ ·α 6= 0 (the latter is because P
the multigraded component of ξ ·α corresponding to C
is ξ · rst [us vt ] 6= 0). Take β = I⊂[m]\{k,l} βI ∈ Lkl \ {0} and choose I ⊂ [m] \ {k, l}such that (β · α)I∪{k,l} = βI · rkl [uk vl ] 6= 0, as in the beginning of the proof. The
multigrading of ξ does not contain l, so we P
have (ξ · α)I∪{k,l} = ξ · rjl [uj vl ] for somej ∈ [m]. Now, ξ · rjl [uj vl ] = 0 because ξ = i∈L1 ±[uJi vi ] and vi vl = 0 for any i ∈L1 , as i and l are not joined by an edge. Hence, ((β+ξ)·α)I∪{k,l} = (β·α)I∪{k,l} 6= 0.
Thus, (β + ξ) · α 6= 0 and we have proved that (Lkl ⊕ hξi) ∩ AnnR α = {0}. Thisimplies that dim AnnR [uk vl ] > dim AnnR α.
32

V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK

Proof of Lemma 4.11. We are given a 3-polytope P from the Pogorelov class P and∼=a ring isomorphism ψ : R = H ∗ (ZP ) −→ H ∗ (ZP 0 ) = R0 . We defined the set
T (P ) = {±[ui vj ] ∈ H 3 (ZP ),Fi ∩ Fj = ∅},0

and the corresponding set for P ,T (P 0 ) = {±[u0i vj0 ] ∈ H 3 (ZP 0 ),Fi0 ∩ Fj0 = ∅}.

We need to show that ψ(T (P )) = T (P 0 ), in other words, ψ([up vq ]) = ±[u0r vs0 ]. We
first use Theorems 4.8 and 4.10 to conclude
that P 0 also belongs to the class P.
P0
0 0
Now suppose that ψ([up vq ]) = α =rij [ui vj ] with at least two nonzero rij . We
are then in the situation of Lemma C.1, which we can apply to P 0 . We obtainthat dim AnnR0 α0 < dim AnnR [u0k vl0 ] for any nonzero summand rkl [u0k vl0 ] of α0 .
−100 0Considering the inverse
P isomorphism ψ : R → R, we can choose [uk vl ] such that−10 0
ψ ([uk vl ]) = α =rab [ua vb ] where [up vq ] appears in the latter sum. As an
isomorphism preserves the dimension of the annihilator subspace, we obtaindim AnnR [up vq ] = dim AnnR0 α0 < dim AnnR [u0k vl0 ] = dim AnnR α< dim AnnR [up vq ],which is a contradiction. It follows that ψ([up vq ]) is a multiple of a single [u0r vs0 ].
Since ψ is an isomorphism over Z, we have ψ([up vq ]) = ±[u0r vs0 ].
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V. BUCHSTABER, N. EROKHOVETS, M. MASUDA, T. PANOV, AND S. PARK[46] Ziegler, Günter M. Lectures on polytopes. Graduate Texts in Mathematics, 152. SpringerVerlag, New York, 1995.
Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina str. 8, 119991
Moscow, RUSSIA,
Department of Mathematics and Mechanics, Moscow State University, andInstitute for Information Transmission Problems, Russian Academy of Sciences
E-mail address: buchstab@mi.ras.ru
Department of Mathematics and Mechanics, Moscow State University, Leninskie
Gory, 119991 Moscow, RUSSIA
E-mail address: erochovetsn@hotmail.com
Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku,
Osaka-shi 558-8585, JAPANE-mail address: masuda@sci.osaka-cu.ac.jp
Department of Mathematics and Mechanics, Moscow State University, Leninskie
Gory, 119991 Moscow, RUSSIA,
Institute for Theoretical and Experimental Physics, Moscow, andInstitute for Information Transmission Problems, Russian Academy of Sciences
E-mail address: tpanov@mech.math.msu.su
URL: http://higeom.math.msu.su/people/taras/
Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138 Sugimoto, Sumiyoshi-ku Osaka-shi 558-8585, JAPANE-mail address: seonjeong1124@gmail.com