aa r X i v : . [ m a t h . A T ] A ug B -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES NIKOLAI EROKHOVETS
Abstract.
Toric topology assigns to each n -dimensional combinatorial simple convex polytope P with m facets an ( m + n )-dimensional moment-angle manifold Z P with an action of a compacttorus T m such that Z P /T m is a convex polytope of combinatorial type P . A simple n -polytopeis called B -rigid, if any isomorphism of graded rings H ∗ ( Z P , Z ) = H ∗ ( Z Q , Z ) for a simple n -polytope Q implies that P and Q are combinatorially equivalent. An ideal almost Pogorelovpolytope is a combinatorial 3-polytope obtained by cutting off all the ideal vertices of an idealright-angled polytope in the Lobachevsky (hyperbolic) space L . These polytopes are exactlythe polytopes obtained from any, not necessarily simple, convex 3-polytopes by cutting off allthe vertices followed by cutting off all the ”old” edges. The boundary of the dual polytope is thebarycentric subdivision of the boundary of the old polytope (and also of its dual polytope). Weprove that any ideal almost Pogorelov polytope is B -rigid. This produces three cohomologicallyrigid families of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds,canonical 6-dimensional quasitoric manifolds and canonical 3-dimensional small covers, whichare ”pullbacks from the linear model”. Introduction
For an introduction to the polytope theory we recommend [Z14]. For brevity, by an n -polytope we call a convex n -dimensional polytope. All the details concerning basic notions from torictopology can be found in [BP15] and [BE17S]. Here we give a brief description of these notions.Let S = { z ∈ C : | z | = 1 } , and T m = ( S ) m be a compact torus. Toric topology assigns to eachsimple n -polytope P with m faces F , . . . , F m an ( m + n )-dimensional moment-angle manifold Z P = T m × P n / ∼ , where ( t , p ) ∼ ( t , p ) if and only if p = p , and t t − ∈ T G ( p ) , and T G ( p ) = { ( t , . . . , t m ) ∈ T m : t i = 1 for F i p } . There is an action of T m on Z P inducedfrom its action on the first factor, and Z P /T m = P . It can be shown that topological type of Z P depends only on combinatorial type of P and Z P has a smooth structure such that theaction of T m is smooth.The mapping Λ : { F , . . . , F m } → Z n such that for any vertex v = F i ∩ · · · ∩ F i n the vectorsΛ( F i ), . . . , Λ( F i n ) form a basis in Z n is called characteristic . It induces a mapping ϕ Λ : T m → T n , ϕ Λ ( t , . . . , t m ) = ( t Λ , t Λ , . . . t Λ ,m m , . . . , t Λ n, t Λ n, . . . t Λ n,m m ) , Mathematics Subject Classification.
Key words and phrases.
Ideal right-angled polytope, B -rigidity, cohomological rigidity, almost Pogorelovpolytope, pullback from the linear model. where S = { z ∈ C : | z | = 1 } , and Λ( F i ) = (Λ ,i , . . . , Λ n,i ) ∈ Z n . It can be shown that thesubgroup Ker ϕ Λ acts on Z P freely. The quotient space is known as a quasitoric manifold : M ( P, Λ) = T n × P n / ∼ , where ( t , p ) ∼ ( t , p ) if and only if p = p , and t t − ∈ ϕ Λ ( T G ( p ) ). The quasitoric manifoldhas an action of T n = T m / Ker ϕ Λ , and M ( P, Λ) /T n = P . These manifolds first appeared in[DJ91] as topological analogs of toric manifolds from algebraic geometry.Similarly, for a Z -characteristic mapping Λ : { F , . . . , F m } → Z n = ( Z / Z ) n such that forany vertex v = F i ∩ · · · ∩ F i n the vectors Λ ( F i ), . . . , Λ ( F i n ) form a basis in Z n there is a small cover R ( P, Λ ) = Z n × P n / ∼ , where ( s , p ) ∼ ( s , p ) if and only if p = p , and s − s ∈ h Λ ( F i ) i F i ∋ p . It can be shown thatany small cover is a smooth n -dimensional manifold glued from 2 n copies of P . Moreover, if P and Q are combinatorially equivalent, then the corresponding small covers are diffeomorphic.The small cover has a canonical action of Z n such that R ( P, Λ ) / Z n = P . Definition 0.1.
A simple n -polytope P is called B -rigid , if for any simple n -polytope Q anyisomorphism of graded rings H ∗ ( Z P , Z ) ≃ H ∗ ( Z Q , Z ) implies that P and Q are combinatoriallyequivalent.A simple n -polytope P is called C -rigid , if it admits a characteristic mapping, and for anysimple n -polytope Q any isomorphism of graded rings H ∗ ( M ( P, Λ) , Z ) ≃ H ∗ ( M ( Q, Λ ′ ) , Z )implies that P and Q are combinatorially equivalent.It is known that B -rigid polytope is C -rigid [CPS10] (see also [BEMPP17]). The converse isnot true [CP19].In what follows, unless otherwise specified, by a polytope we mean a class of combinatoriallyequivalent 3-polytopes. By faces we call its facets.For any simple 3-polytope P by the Four colour theorem there is a colouring c : { F , . . . , F m } →{ , , , } such that adjacent faces have different colours. Each colouring c induces a character-istic function Λ c by the following rule. Let e , e , and e be a basis in Z , and e = e + e + e .Set Λ c ( F i ) = e c ( i ) . Then Λ c is a characteristic function, since any three vectors from the set { e , e , e , e } form a basis in Z . Definition 0.2. A k -belt is a cyclic sequence of k faces with the property that faces are adjacentif and only if they follow each other, and no three faces have a common vertex. A k -belt is trivial , if it surrounds a face. Definition 0.3.
A simple n -polytope P is called flag , if any set of its pairwise intersectingfaces F i , . . . , F i k has a non-empty intersection F i ∩ · · · ∩ F i k = ∅ .It can be shown (see [BE15]) that a simple 3-polytope P is flag if and only if P = ∆ , and P has no 3-belts.Results by A.V. Pogorelov [P67] and E.M. Andreev [A70a] imply that a simple 3-polytope P can be realized in the Lobachevsky (hyperbolic) space L as a bounded polytope with right -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 3 dihedral angles if and only if P is different from the simplex ∆ and has no 3- and 4-belts.Moreover, a realization is unique up to isometries. Such polytopes are called Pogorelov polytopes .We denote their family P P og . It follows from [D98, D03] (see also [BE17I]) that P P og contains fullerenes , that is simple 3-polytopes with only 5- and 6-gonal. Mathematical fullerenes modelspherical carbon atoms. In 1996 R. Curl, H. Kroto, and R. Smalley obtained the Nobel Prize inchemistry “for their discovery of fullerenes”. They synthesized
Buckminsterfullerene C , whichhas the form of the truncated icosahedron (and also of a soccer ball). W.P. Thurston [T98] builta parametrisation of the fullerene family, which implies that the number of fullerenes with n carbon atoms grows like n when n tends to infinity.In [FMW15] F. Fan, J. Ma, and X. Wang proved that any Pogorelov polytope is B -rigid. Inparticular, it is C -rigid. Remark . The notions of B - and C -rigidity can be defined for any field F instead of Z . Thena polytope B -rigid over F is C -rigid over F . In fact, in paper [FMW15] it is proved that anyPogorelov polytope is B -rigid over Z or any field F . Remark . In [B17] F. Bosio presented a construction of flag 3-polytopes, which are not B -rigid.In [BEMPP17] for Pogorelov polytopes a fact stronger than C -rigidity was proved. Namely,it is known (see [DJ91, BP15, BEMPP17]) that the manifolds M ( P, Λ) and M ( Q, Λ ′ ) overtwo simple n -polytopes P and Q are weakly equivariantly diffeomorphic, that is there is adiffeomorphism f : M ( P, Λ) → M ( Q, Λ ′ ) and an automorphism ψ : T n → T n such that f ( t · x ) = ψ ( t ) · f ( x ) for all t ∈ T n and x ∈ M ( P, Λ), if and only if there is a combinatorial equivalence ϕ : P → Q and a change of basis C ∈ GL ( n, Z ) such that Λ( ϕ ( F i )) = ± C · Λ( F i ) for i = 1 , . . . , m .Such pairs ( P, Λ) and ( Q, Λ ′ ) are called equivalent . Theorem 0.4 ([BEMPP17]) . Let
P, Q ∈ P
P og . Then there is an isomorphism of graded rings H ∗ ( M ( P, Λ) , Z ) ≃ H ∗ ( M ( Q, Λ ′ ) , Z ) if and only if the pairs ( P, Λ) and ( Q, Λ ′ ) are equivalent. Moreover, in [BP16] it was proved that the pairs ( P, Λ c ) and ( Q, Λ ′ c ′ ) induced by colouringsare equivalent if and only if there is a combinatorial equivalence ϕ : P → Q and a permutation σ ∈ S such that c ′ ( ϕ ( F i )) = σ ◦ c ( F i ) for all i .A similar result was proved for small covers. Namely, any characteristic function Λ inducesa Z -characteristic function Λ simply by taking all the coefficients modulo 2. In dimension 3the converse is also true. The inclusion of sets Z ⊂ Z lifts a Z -characteristic function Λ toa characteristic function Λ, since for any matrix of size 3 × Z , the determinant is ± Z . This is not valid for dimensions greaterthan 3. It is known [DJ91] that H ∗ ( R ( P, Λ ) , Z ) = Z [ v , . . . , v m ] / ( I P + J Λ ) , where each variable v i has degree 1, the ideal I P is generated by monomials v i . . . v i k such that F i ∩ · · · ∩ F i k = ∅ , and the ideal J Λ is generated by linear forms Λ , v + · · · + Λ ,m v m , . . . ,Λ n, v + · · · + Λ n,m v m . Similarly for a quasitoric manifold M ( P, Λ), and the coefficient ring R ,which is Z or Z we have [DJ91]: H ∗ ( M ( P, Λ) , R ) = R [ v , . . . , v m ] / ( I P + J Λ ) , N.YU. EROKHOVETS where each variable v i has degree 2.Thus, any isomorphism of graded rings ϕ : H ∗ ( R ( P, Λ ) , Z ) → H ∗ ( R ( Q, Λ ′ ) , Z ) inducesan isomorphism for M ( P, Λ) and M ( Q, Λ ′ ). In particular, if P and Q are Pogorelov polytopes,then they are combinatorially equivalent. Moreover, it was proved in [BEMPP17] that the pairs( P, Λ ) and ( Q, Λ ′ ) in this case are Z -equivalent , that is there is a a combinatorial equivalence ϕ : P → Q and a change of basis C ∈ GL (3 , Z ) such that Λ( ϕ ( F i )) = C · Λ( F i ) for i = 1 , . . . , m .Recent results of this type in the context of toric varieties for n -dimensional cubes see in[CLMP20]. Construction 0.5.
In [V87] A.Yu. Vesnin introduced a construction of a 3-dimensionalcompact hyperbolic manifold N ( P, Λ ) corresponding to a Pogorelov polytope P and a Z -characteristic mapping Λ . Namely, for a bounded right-angled polytope P ⊂ L there is aright-angled Coxeter group G ( P ) generated by reflections in faces of P . It it known that G ( P ) = h ρ , . . . , ρ m i / ( ρ , . . . , ρ m , ρ i ρ j = ρ j ρ i for all F i ∩ F j = ∅ ) , where ρ i is the reflection in the face F i . The group G ( P ) acts on L discretely (see detailsin [VS88]), and P is a fundamental domain of G ( P ), that is the interiors of the polytopes { gP } g ∈ G ( P ) do not intersect. Moreover, the stabiliser of a point x ∈ P is generated by reflectionsin faces containing it.The mapping Λ defines a homomorphism ϕ : G ( P ) → Z by the rule ϕ ( ρ i ) = Λ ( F i ). It canbe shown that Ker ϕ acts on L freely. Then N ( P, Λ ) = L / Ker ϕ is a compact hyperbolicmanifold glued of 8 copies of P along faces. It is easy to see that N ( P, Λ ) is homeomorphic toa small cover R ( P, Λ ), and the homeomorphism is given by the mapping ( t, p ) → ϕ − ( t ) · p .In this paper we consider another family P aP og of 3-polytopes. It consists of simple 3-polytopes P = ∆ without 3-belts such that any 4-belt surrounds a face. We call them almost Pogorelovpolytopes . The combinatorics and hyperbolic geometry of the family P aP og was studied in [E19].In 2019 T.E. Panov remarked that results by E. M. Andreev [A70a, A70b] should imply thatalmost Pogorelov polytopes correspond to right-angled polytopes of finite volume in L . Suchpolytopes may have 4-valent vertices on the absolute, while all proper vertices have valency3. It was proved in [E19] that cutting of 4-valent vertices defines a bijection between classesof congruence of right-angled polytopes of finite volume in L and almost Pogorelov polytopesdifferent from the cube I and the pentagonal prism M × I . Moreover, it induces the bijectionbetween the ideal vertices of the right-angled polytope and the quadrangles of the correspond-ing almost Pogorelov polytope. The polytopes I and M × I are the only almost Pogorelovpolytopes with adjacent quadrangles. The cube has m = 6 faces. The pentagonal prism has7 faces. For m = 8 there are no almost Pogorelov polytopes. For m = 9 there is a uniquealmost Pogorelov polytope – the 3-dimensional associahedron As . It can be realized as a cubewith three non-adjacent pairwise orthogonal edges cut. It follows from [B74] that all the otherpolytopes in P aP og can be obtained from As by a sequence of operations of cutting off an edgenot lying in quadrangles, and cutting off a pair of adjacent edges of a face with at least 6 sidesby one plane. Moreover, these operations preserve the family of almost Pogorelov polytopes. -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 5 a) b) k-gon Figure 1. a) Medial graph of the k -gonal pyramid; b) The k -antiprism Definition 0.6.
A polytope in L is called ideal , if all its vertices lie on the absolute (are idealpoints ). An ideal polytope has a finite volume. Its congruence class is uniquely defined by thecombinatorial type [R94] (see also [R96]). We will call almost Pogorelov polytopes correspondingto ideal right-angled polytopes ideal almost Pogorelov polytopes and denote their family P IP og .It is a classical fact (see, for example [T02, Section 13.6] and [BGGMTW05, Theorem 5 (iv)])that graphs of ideal right-angled 3-polytopes are exactly medial graphs of 3-polytopes. Verticesof a medial graph correspond to edges of a polytope, and edges – to pairs of edges adjacentin a face. This correspondence plays a fundamental role in the well-known Koebe-Andreev-Thurston theorem (see [Z14, T02, BS04, S92]): any -dimensional combinatorial polytope P has a geometric realization in the Euclidean space R such that all its edges are tangent toa given sphere . A medial graph of the k -pyramid is known as k -antiprism, see Fig. 1 b). Itfollows from the definition that ideal almost Pogorelov polytope P corresponding to a polytope Q can be obtained from it by cutting off all the vertices followed by cutting off all the ”old”edges. Such polytopes appeared in [DJ91, Example 1.15(1)] as polytopes giving examples forquasitoric manifolds and small covers, which are ”pullbacks from the linear model”. We provethat for 3-polytopes the families of these manifolds are cohomologically rigid (see Theorem8.2). The triangulation ∂P ∗ is the barycentric subdivision of ∂Q ∗ . Such triangulations appearedalso in [FMW20, Example 6.3(2)] as examples of flag triangulations such that any 4-circuit fullsubcomplex is simple. Also they appeared in [CK11] in study of combinatorial rigidity (a simple n -polytope P is called combinatorially rigid , if there are no other simple n -polytopes with thesame bigraded Betti numbers β − i, j ( Z P )).An operation of an edge-twist is drawn on Fig. 2. Two edges on the left lie in the same faceand are disjoint. Let us call an edge-twist restricted , if both edges are adjacent to an edge ofthe same face. It follows from [V17, BGGMTW05, E19] that a polytope is realizable as an idealright-angled polytope if and only if either it is a k -antiprism, k >
3, or it can be obtained fromthe 4-antiprism by a sequence of restricted edge-twists. Thus, the simplest ideal right-angledpolytope is a 3-antiprism, which coincides with the octahedron. The corresponding ideal almostPogorelov polytope is known as 3-dimensional permutohedron
P e , see Fig. 3. This is a uniqueideal almost Pogorelov polytope with minimal number m of faces (14 faces). It is easy to see N.YU. EROKHOVETS
Figure 2.
An edge-twist
Figure 3.
Three-dimensional permutohedron
P e that for ideal almost Pogorelov polytopes m = 2( p + 1), where p is the number of quadrangles.Recent results on volumes of ideal right-angled polytopes see in [VE20].In the paper [E20] the technique from [FMW15] (see also [BE17S]) was generalized to thefamily P aP og . In particular, it was proved that if there is an isomorphism of graded rings H ∗ ( Z P , Z ) ≃ H ∗ ( Z Q , Z ), where P is an almost Pogorelov or an ideal almost Pogorelov polytope,and Q is any simple 3-polytope, then Q is an almost Pogorelov [E20, Theorem 3.6] or an idealalmost Pogorelov polytope [E20, Corollary 3.8] respectively.Main result of our paper is Theorem 0.7.
Any ideal almost Pogorelov polytope is B -rigid. The proof follows from results in [E20]. We compare it with the proof of B -rigidity ofPogorelov polytopes from [FMW15].In Theorem 8.2 we show how Theorem 0.7 gives three cohomologically rigid families of man-ifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canonical 6-dimensionalquasitoric manifolds and canonical 3-dimensional small covers, which are ”pullbacks from thelinear model” from [DJ91, Example 1.15(1)].Below is the plan of the paper.In Section 1 we give a brief description of the cohomology ring of a moment-angle manifold.In Section 2 we describe an important tool in the study of families if 3-polytopes: a separablecircuit condition.In Section 3 we describe a notion of a B -rigid property, subset, a collection of subsets, or anumber. We explain, why the properties to be a flag, Pogorelov and almost Pogorelov polytopesare B -rigid. Also there are several interesting open problems in this section.In Section 4 we describe B -rigid subsets in H ( Z P ) for Pogorelov and almost Pogorelovpolytopes. -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 7 In Section 5 we explain, which sets of elements in H k +2 ( Z P ) corresponding to k -belts are B -rigid for Pogorelov and almost Pogorelov polytopesIn Section 6 we deal with belts surrounding faces. It turns out that for Pogorelov polytopesor ideal almost Pogorelov polytopes P, Q the elements in H ∗ ( Z P ) corresponding to belts aroundfaces under each isomorphism of graded rings H ∗ ( Z P ) → H ∗ ( Z Q ) are mapped to the analogouselements for Q . This induces a bijection between the sets of faces of P and Q .In Section 7 we discuss that under this bijection adjacent faces are mapped to adjacent faces.In particular, there is a combinatorial equivalence between P and Q . This proves Theorem 0.7.In Section 8 we show (Theorem 8.2) how Theorem 0.7 produces three cohomologically rigidfamilies of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canon-ical 6-dimensional quasitoric manifolds and canonical 3-dimensional small covers, which are”pullbacks from the linear model” from [DJ91, Example 1.15(1)].In Section 9 we study hyperbolic geometry of the 3-dimensional manifolds corresponding toideal almost Pogorelov polytopes. Remark . In [FMW20, Theorems 3.12 and 3.14] F. Fan, J. Ma, and X. Wang generalizedresults from [FMW15] and [BEMPP17] to flag polytopes of dimension higher than 3 satisfyinga generalization of the separable circuit condition (see Section 2). It is announced withoutany details, that in the next paper [FMW20b] the authors will prove B -rigidity of any almostPogorelov polytope and an analog of Theorem 0.4 for them. But their [FMW20, Definition 2.16]of B -rigidity assumes isomorphism of bigraded rings, which is a more restrictive condition.1. Cohomology ring of a moment-angle manifold of a simple -polytope Details on cohomology of a moment-angle manifolds see in [BP15, BE17S]. If not specified,we study cohomology over Z and omit the coefficient ring.The ring H ∗ ( Z P ) has a multigraded structure: H ∗ ( Z P ) = M i > ,ω ⊂ [ m ] H − i, ω ( Z P ) , where H − i, ω ( Z P ) ⊂ H | ω |− i ( Z P ) , and [ m ] = { , . . . , m } . There is a canonical isomorphism H − i, ω ( Z P ) ≃ e H | ω |− i − ( P ω ), where P ω = S i ∈ ω F i , and e H − ( ∅ ) = Z .The multiplication between components is nonzero, only if ω ∩ ω = ∅ . The mapping H − i, ω ( Z P ) ⊗ H − j, ω ( Z P ) → H − ( i + j ) , ω ⊔ ω ) ( Z P )via the Poincare-Lefschets duality H i ( P ω ) ≃ H n − − i ( P ω , ∂P ω ) up to signs is induced by inter-section of faces of P .For any simple 3-polytope P and ω = ∅ the set P ω is a 2-dimensional manifold, perhaps witha boundary. Therefore, e H k ( P ω ) is nonzero only for ( k, ω ) ∈ { ( − , ∅ ) , (0 , ∗ ) , (1 , ∗ ) , (2 , [ m ]) } . Inparticular, P has no torsion, and there is an isomorphism H ∗ ( Z P , Q ) ≃ H ∗ ( Z P ) ⊗ Q and theembedding H ∗ ( Z P ) ⊂ H ∗ ( Z P ) ⊗ Q . For polytopes P and Q the isomorphism H ∗ ( Z P ) ≃ H ∗ ( Z Q )implies the isomorphism over Q . For cohomology over Q or any field F all theorems about thestructure of H ∗ ( Z P , Q ) are still valid. N.YU. EROKHOVETS
There is a multigraded Poincare duality, which means that the bilinear form H − i, ω ( Z P ) × H − ( m − − i ) , m ] \ ω ) ( Z P ) → H − ( m − , m ] ( Z P ) = Z has in some basis matrix with determinant ±
1. We have H ( Z P ) = e H − ( ∅ ) = Z = e H ( ∂P ) = H m +3 ( Z P ); H ( Z P ) = H ( Z P ) = 0 = H m +1 ( Z P ) = H m +2 ( Z P ); H k ( Z P ) = M | ω | = k − e H ( P ω ) ⊕ M | ω | = k − e H ( P ω ) , k m. Nontrivial multiplication occurs only for(1) e H − ( P ∅ ) ⊗ e H k ( P ω ) → e H k ( P ω ). This corresponds to multiplication by 1 in H ∗ ( Z P ).(2) e H ( P ω ) ⊗ e H ( P [ m ] \ ω ) → e H ( ∂P ). This corresponds to the Poincare duality pairing.(3) e H ( P ω ) ⊗ e H ( P ω ) → e H ( P ω ⊔ ω ).A crucial role in B -rigidity questions for 3-polytopes is played by the following two types ofclasses in H ∗ ( Z P ). Denote N ( P ) = {{ i, j } ⊂ [ m ] : F i ∩ F j = ∅ } . For any ω ∈ N ( P ) we have e H ( P ω ) = Z . Choose a generator in this group and denote it by e ω .Then for any 3-polytope H ( Z P ) is a free abelian group with the basis { e ω : ω ∈ N ( P ) } .For any k -belt B k we have e H ( P ω ) = Z for ω = { i : F i ∈ B k } . Choose a generator in thisgroup and denote it by f B k . Denote by B k the subgroup in H k +2 ( Z P ), 3 k m −
2, with thebasis { f B k : B k is a k -belt } .2. Separable circuit condition for families of -polytopes A very important tools for study of families of 3-dimensional polytopes are given by so-calledseparable circuit conditions (SCC for short) [FMW20].
Proposition 2.1 (SCC for flag polytopes [FW15]) . A simple -polytope is flag if and only iffor any three pairwise different faces { F i , F j , F k } with F i ∩ F j = ∅ there exists an l -belt B l suchthat F i , F j ∈ B l and F k / ∈ B l . Denote |B l | = S F i ∈B l F i . Proposition 2.2 (SCC for Pogorelov polytopes [FMW15]) . A simple -polytope is Pogorelovif and only if for any three pairwise different faces { F i , F j , F k } with F i ∩ F j = ∅ there existsan l -belt B l such that F i , F j ∈ B l , F k / ∈ B l , and F k does not intersect at least one of the twoconnected components of |B l | \ ( F i ∪ F j ) .Remark . In [FMW20] SCC for Pogorelov polytopes was generalized to higher dimensions (wecall it SCC’ for short). In particular, the product of two flag polytopes with SCC’ is also a flagpolytope with SCC’. For a flag simple n -polytope with SCC’ there is a construction [FMW20,Construction E.1] which under some assumptions gives a flag ( n + 1)-polytope with SCC’. -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 9 Let P be the cube with two non-adjacent orthogonal edges cut. Proposition 2.3 (SCC for almost Pogorelov polytopes [FMW20, E20]) . A simple -polytope P is an almost Pogorelov polytope or the polytope P if and only if for any three pairwise differentfaces { F i , F j , F k } such that F i ∩ F j = ∅ an l -belt B l such that F i , F j ∈ B l , F k / ∈ B l , and F k doesnot intersect at least one of the two connected components of |B l | \ ( F i ∪ F j ) exists if and onlyif F k does not intersect quadrangles among the faces F i and F j .Remark . In [FMW20, Proposition F.1] this result was first proved in the dual setting forsimplicial polytopes in ”only if” direction. In [E20] this result was proved independently forsimple polytopes in both directions.3. B -rigid properties Definition 3.1.
A property of an n -polytope P is called B -rigid , if any isomorphism of gradedrings H ∗ ( Z P , Z ) = H ∗ ( Z Q , Z ) for a simple n -polytope Q implies that it also has this property. Definition 3.2.
Let P be some set of 3-polytopes.We call a number n ( P ), a set S P ⊂ H ∗ ( Z P ) or a collection of such sets defined for anypolytope P ∈ P B -rigid in the class P if for any isomorphism ϕ of graded rings H ∗ ( Z P ) ≃ H ∗ ( Z Q ), P, Q ∈ P , we have n ( P ) = n ( Q ), ϕ ( S P ) = S Q , or each set from the collection for P is mapped bijectively to some set from the collection for Q respectively.To be short, B -rigidity in the class of all simple 3-polytopes we call B -rigidity . Proposition 3.3.
A property to be a flag -polytope is B -rigid. There are at least three different proofs of this fact. First, in [FW15] it was proved that it isequivalent to the fact that the ring e H ∗ ( Z P ) / ([ Z P ]) is a (nonzero) indecomposable ring. Also itwas proved that for flag 3-polytopes e H ( P ω ) = M ω ⊔ ω e H ( P ω ) · e H ( P ω ) . This result was based on SCC for flag polytopes (Proposition 2.1). Using the latter fact in[BE17S] and [BEMPP17] it was proved that a simple 3-polytope P = ∆ is flag if and only if H m − ( Z P ) ⊂ ( e H ∗ ( Z P )) . In [E20, Proposition 2.2] it was proved that a simple 3-polytope P = ∆ with m faces is flag ifand only if rk H ( Z P ) = ( m − m − m − .For simple polytopes of dimension n > Problem . Is the property to be a flag n -polytope B -rigid in the class of simple n -polytopes?Also there is another known problem. Problem . Is dimension n of a simple polytope B -rigid in the class of all simple polytopes? It is easy to see that a polytope has no 4-belts if and only if the multiplication H ( Z P ) ⊗ H ( Z P ) → H ( Z P )is trivial. Thus, the property to be a Pogorelov polytope is also B -rigid. Proposition 3.4. [FMW15] (see also a proof in [BE17S, Proposition 8.24] ) The subgroup B k ⊂ H k +2 ( Z P ) is B -rigid for k m − . In particular, rk B k = { k -belts } for k > isa B -rigid number. In [E20] it was proved that the property to be an almost Pogorelov polytope is B -rigid.Namely, the polytopes I and M × I are B -rigid, since these are unique flag polytopes with m = 6 and m = 7 respectively. Let A ( P ) be a subgroup in H ( Z P ), which is a kernel of thebilinear form defined by the product map H ( Z P ) ⊗ H ( Z P ) → H ( Z P ), that is A = { x ∈ H ( Z P ) : x · y = 0 for all y ∈ H ( Z P ) } . It is generated by elements ^ { p, q } such that the faces F p and F q are not adjacent and do notbelong to any 4-belt B .Denote by I the image of the mapping H ( Z P ) ⊗ H ( Z P ) → H ( Z P ) Theorem 3.5. [E20, Theorem 3.12]
A flag simple -polytope P belongs to P aP og \ { I , M × I } if and only if B = rk (cid:2) H ( Z P ) /A ( P ) (cid:3) , and rk I = rk B + ( m − B . In particular, the property to be an almost Pogorelov polytope is B -rigid. An almost Pogorelov polytope is ideal if and only if any its vertex lies on a unique quadrangle.This is equivalent to the fact that 4 p = f , where f is the number of vertices, and p is thenumber of quadrangles. For any simple 3-polytope we have f = 2( m − P = I , M × I we have p = rk B . Corollary 3.6.
A polytope in P aP og \{ I , M × I } belongs to P IP og if and only if B = m − .In particular, the property to be an ideal almost Pogorelov polytope is B -rigid.Problem . We call by an almost flag polytope a simple 3-polytope such that any its 3-belt istrivial. Is the property to be an almost flag polytope B -rigid?4. Three-dimensional classes
In [FMW15] it was proved that the set of elements {± e ω : ω ∈ N ( P ) } ⊂ H ( Z P )is B -rigid in the class of Pogorelov polytopes.The proof was based on the annihilator lemma . An annihilator of an element r in a ring R is Ann R ( r ) = { s ∈ R : rs = 0 } . -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 11 Lemma 4.1 (annihilator lemma, [FMW15]) . Let P ∈ P P og , and α ∈ H = H ∗ ( Z P , Q ) : α = X ω ∈ N ( P ) r ω e ω with |{ ω : r ω = 0 }| > Then dim Ann H ( α ) < dim Ann H ( e ω ) , if r ω = 0 . For almost Pogorelov polytopes the annihilator lemma is not valid. A counterexample arisesalready for P = As , see [E20, Proposition 8.3]. Definition 4.2.
Let P be a simple 3-polytope. An element ω ′ = { s, t } ∈ N ( P ) is good for anelement ω = { p, q } ∈ N ( P ), if there is an l -belt B l containing F s and F t such that either F p or F q does not belong to B l and does not intersect at least one of the two connected componentsof B l \ { F s , F t } . Lemma 4.3. [E20, Lemma 6.1]
Let P be a simple -polytope and α ∈ H = H ∗ ( Z P , Q ) : α = X ω ∈ N ( P ) r ω e ω with |{ ω : r ω = 0 }| > . Then for any ω = { p, q } with r ω = 0 we have (1) dim Ann H ( α ) dim Ann H ( e ω ) ; (2) dim Ann H ( α ) < dim Ann H ( e ω ) , if there is ω ′ = { s, t } with r ω ′ = ∅ , which is good for ω . Corollary 4.4. [E20, Corollary 6.4]
Let P be a simple -polytope. If any ω ′ ∈ N ( P ) \{ ω } is goodfor an element ω ∈ N ( P ) , then for any isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) for a simple -polytope Q we have ϕ ( e ω ) = ± e ω ′ for some ω ′ ∈ N ( Q ) . It follows from the SCC for an almost Pogorelov polytope P that for ω = { p, q } , where F p and F q are quadrangles, any other element ω ′ ∈ N ( P ) is good. Thus, for any isomorphism ofgraded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) for a simple 3-polytope Q we have ϕ ( e ω ) = ± e ω ′ for some ω ′ ∈ N ( Q ). To prove that ω ′ also corresponds to a pair of quadrangles an additional techniqueis needed.First, for any polytope P ∈ P aP og \ { I , M × I } and any its 4-belt B define a subgroup G ( B ) in H = H ( Z P ) /A generated by two cosets ^ { p, q } + A ( P ) corresponding to pairs ofopposite faces of B . Lemma 4.5. [E20, Lemma 7.1]
The collection of subgroups { G ( B ) : B is a -belt } is B -rigid in the class P aP og \ { I , M × I } . As a corollary the set {± f B } of generators of B is B -rigid in the class P aP og \ { I , M × I } ,since these are generators of the images of the mappings G ( B ) ⊗ G ( B ) → H ( Z P ). Therefore,for P, Q ∈ P aP og \ { I , M × I } any isomorphism of graded rings H ∗ ( Z P ) → H ∗ ( Z Q ) induces abijection ϕ between the sets of quadrangles of P and Q by the rule ϕ ( F i ) = F ′ i ′ , where ϕ ( e B i ) = ± f B ′ i ′ for 4-belts B i and B ′ i ′ around F i and F ′ i ′ . On the base of SCC for almost Pogorelov polytopes the next result was proved.
Lemma 4.6. [E20, Lemma 7.5]
The collection of cosets of the form ± ^ { p, q } + A ( P ) , where F p and F q are opposite faces of a -belt, is B -rigid in the class P aP og \ { I , M × I } . The next important step is the following
Lemma 4.7. [E20, Proposition 7.9]
Let
P, Q ∈ P aP og \ { I , M × I } , and let F p be a quadrangleof P not adjacent to a face F q . Assume that for an isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) we have ϕ ( ^ { p, q } ) = ± ] { s, t } for some { s, t } ∈ N ( Q ) . Then p ′ ∈ { s, t } for F ′ p ′ = ϕ ( F p ) .In particular, ϕ ( ^ { p, q } ) = ± ^ { p ′ , q ′ } for quadrangles F p and F q , and the set {± ^ { p, q } : F p and F q are quadrangles } ⊂ H ( Z P ) is B -rigid in the class P aP og \ { I , M × I } . The proof is based on the following combinatorial fact and lemmas.
Lemma 4.8. [E20, Lemma 7.11]
Let P ∈ P aP og \ { I , M × I } . Then for any three pairwisedifferent faces { F i , F j , F k } such that F i ∩ F j = ∅ , F k is a quadrangle, and at least one of thefaces F i and F j is not adjacent to F k , there exists an l -belt B l such that F i , F j ∈ B l , F k / ∈ B l ,and B l does not contain any of the two pairs of opposite faces of the -belt around F k . Lemma 4.9. [E20, Lemma 7.8]
Let P ∈ P aP og \ { I , M × I } , and let B k be a k -belt passingthrough a quadrangle F i and its adjacent faces F p an F q . Let x be a generator of e H ( P τ ) = Z for τ = ω ( B k ) (in this case x = ± f B k ), or τ = ω ( B k ) ⊔ { r } , where F r either is not adjacent tofaces in B k , or is adjacent to exactly one face in B k . Then x is divisible by any element in thecoset ^ { p, q } + A ( P ) . Lemma 4.10. [E20, Corollary 7.7]
Let
P, Q ∈ P aP og \ { I , M × I } , and let B k be a k -beltpassing through a quadrangle F i and its adjacent faces F p and F q . Then for any isomorphismof graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) we have ϕ ( f B k ) = X j µ j g B ′ k,j for k -belts B ′ k,j of Q such that for any µ j = 0 the belt g B ′ k,j passes through the non-adjacentfaces F ′ p ′ and F ′ q ′ of Q , where ϕ ( ^ { p, q } + A ( P )) = ± ^ { p ′ , q ′ } + A ( Q ) . Moreover, F ′ p ′ and F ′ q ′ areadjacent to the quadrangle F ′ i ′ = ϕ ( F i ) of Q . -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 13 Classes corresponding to belts
On the base of the B -rigid set of elements corresponding to pairs of non-adjacent faces in[FMW15] it was proved that the set of generators of B k : {± f B k : B k − a k -belt } ⊂ H k +2 ( Z P )is B -rigid in the class of Pogorelov polytopes.As we mentioned above for the class P aP og \ { I , M × I } this result is valid for k = 4.Also it can be proved [E20, Corollary 6.8] that if P ∈ P aP og \ { I , M × I } and a k -belt B k does not have common points with quadrangles, then under any isomorphism of graded rings H ∗ ( Z P ) → H ∗ ( Z Q ) for a simple 3-polytope Q the element f B k is mapped to ± f B ′ k for some k -belt B ′ k of Q .For an ideal almost Pogorelov polytope any face, which is not a quadrangle, is surroundedby a (2 k )-belt containing k quadrangles, k > Lemma 5.1. [E20, Lemma 7.12]
Let
P, Q ∈ P aP og \ { I , M × I } , and let B k be a (2 k ) -belt of P containing k quadrangles. Then for any isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) we have ϕ ( f B k ) = ± f B ′ k for a (2 k ) -belt B ′ k in Q containing k quadrangles. In particular, theset of elements {± f B k : B k − a (2 k ) -belt containing k quadrangles } ⊂ H k +2 ( Z P ) is B -rigid in the class P aP og \ { I , M × I } . The proof is based on Lemmas 4.7 and 4.10, and
Lemma 5.2. [E20, Corollary 6.10]
Let P be a simple -polytope and ω = { p, q } ∈ N ( P ) . Thenan element x = P j µ j g B k,j ∈ B k is divisible by e ω if and only if each g B k,j with µ j = 0 is divisibleby e ω , and if and only if each belt B k,j with µ j = 0 contains F p and F q . Classes corresponding to trivial belts
Then in [FMW20] it was proved that the set of elements {± f B k : B k − a k -belt around a face } ⊂ H k +2 ( Z P )is B -rigid in the class of Pogorelov polytopes. This induces a bijection between the sets of facesof polytopes.We have an analog of this property. Lemma 6.1. [E20, Lemma 7.13]
Let
P, Q ∈ P aP og \ { I , M × I } , and let B k be a trivial (2 k ) -belt of P containing k quadrangles. Then for any isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) we have ϕ ( f B k ) = ± f B ′ k for a trivial (2 k ) -belt B ′ k in Q containing k quadrangles. Inparticular, the set of elements {± f B k : B k − a trivial (2 k ) -belt containing k quadrangles } ⊂ H k +2 ( Z P ) is B -rigid in the class P aP og \ { I , M × I } . The idea of the proof is similar to the idea of the proof for Pogorelov polytopes from [FMW15],namely, we consider elements in H k +3 ( Z P ) having common divisors of a proper form with f B k .For Pogorelov polytopes it is sufficient to consider elements e ω , ω ∈ N ( P ). But for almostPogorelov polytopes we need to consider classes corresponding to pairs of quadrangles, andcosets of the form ± ^ { p, q } + A ( P ) corresponding to faces F p and F q adjacent to a quadranglein the belt.For ideal almost Pogorelov polytopes Lemma 6.1 induces a bijection between the sets offaces of P and Q , which are not quadrangles. Together with ϕ this bijections form a bijectionbetween the sets of faces of P and Q , since any trivial belt of a polytope P ∈ P aP og \{ I , M × I } surrounds exactly one face (for otherwise, P is a prism and has adjacent quadrangles).7. Belts around adjacent faces
Then in [FMW15] it was proved that for Pogorelov polytopes the images of adjacent facesunder the mapping induced by the isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) areadjacent. This follows from the fact that the trivial belts B and B surround adjacent faces ifand only if f B and f B have exactly one common divisor among e ω , ω ∈ N ( P ). This finishes theproof of B -rigidity of any Pogorelov polytope.For ideal almost Pogorelov polytopes we have analogous facts. Lemma 7.1. [E20, Lemma 7.14]
Let P ∈ P aP og \ { I , M × I } , F i be its quadrangle, and let B k be the (2 k ) -belt containing k quadrangles and surrounding a face F j . Then F i and F j areadjacent if and only if the element f B k is divisible by any element in the coset ^ { p, q } + A ( P ) for one of the two pairs of opposite faces { F p , F q } of the -belt B i around F i . In particular, if F i and F j are adjacent, then for any isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) fora polytope Q ∈ P aP og \ { I , M × I } we have ϕ ( f B k ) = ± f B ′ k for a (2 k ) -belt B ′ k containing k quadrangles and surrounding a face F ′ j ′ adjacent to F ′ i ′ = ϕ ( F i ) . Lemma 7.2. [E20, Lemma 7.15]
Let P ∈ P aP og \ { I , M × I } , and let B k be the (2 k ) -belt containing k quadrangles and surrounding a face F i , and B l be the (2 l ) -belt containing l quadrangles and surrounding a face F j . Then F i and F j are adjacent if and only if the elements f B k and f B l have exactly one common divisor among elements ^ { p, q } corresponding to pairs ofquadrangles. In particular, if F i and F j are adjacent, then for any isomorphism of graded rings ϕ : H ∗ ( Z P ) → H ∗ ( Z Q ) for a polytope Q ∈ P aP og \ { I , M × I } we have ϕ ( f B k ) = ± f B ′ k fora (2 k ) -belt B ′ k containing k quadrangles and surrounding a face F ′ i ′ , and ϕ ( f B l ) = ± f B ′ l for a (2 l ) -belt B ′ l containing l quadrangles and surrounding a face F ′ j ′ adjacent to F ′ i ′ . This finishes the proof of B -rigidity of any ideal almost Pogorelov polytope. -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 15 New cohomologically rigid families of manifolds
Definition 8.1.
A family M of manifolds is called cohomologically rigid over the ring R if twomanifolds M , M ∈ M are diffeomorphic if and only if there is an isomorphism of graded rings H ∗ ( M , R ) ≃ H ∗ ( M , R ). B -rigidity of any Pogorelov polytope gives rise to cohomologically rigid over Z family ofmoment-angle manifolds. Moreover, all the theory in [FMW15] works over any field F , thereforethis family is cohomologically rigid over any field F .If we choose for any Pogorelov polytope P some quasitoric manifold M ( P ), then the family { M ( P ) } of 6-dimensional manifolds is cohomologically rigid over Z or any field F . The prob-lem is that there is no canonical way to choose such a manifold. Also, if we choose for eachPogorelov polytope a small cover R ( P ), then the family of 3-dimensional manifolds { R ( P ) } iscohomologically rigid over Z .Theorem 0.4 implies that the families of 6-dimensional quasitoric manifolds and 3-dimensionalsmall covers corresponding to the same Pogorelov polytope P are cohomologically rigid over Z and Z respectively.For ideal almost Pogorelov polytopes we also obtain a cohomologically rigid family ofmoment-angle manifolds. For a polytope P with m faces (recall that m = 2( p + 1), where p is the number of quadrangles) the manifold Z P has dimension m + 3 = 2 p + 5.For ideal almost Pogorelov polytopes the advantage is that there is a canonical way to choosea characteristic function. Namely, it is known that a 3-polytope can be coloured in 3 colourssuch that adjacent faces have different colours if and only if any its face has an even numberof edges (see [I01, J01]). Moreover, such a colouring is unique up to a permutation of colours.This gives a canonical characteristic function Λ P for P up to a permutation of basis vectors in Z (this function appears already in [DJ91, Example 1.15(1)]). Since a change of basis in Z and Z does not change the (weak equivariant) diffeomorphic type of manifolds M ( P, Λ) and R ( P, Λ ), for any ideal almost Pogorelov polytope P there is a canonical quasitoric manifold M ( P ) and a canonical small cover R ( P ). We obtain three families of manifolds.(1) The family Z IP og of moment-angle manifolds corresponding to ideal almost Pogorelovpolytopes.(2) The family M IP og of canonical 6-dimensional quasitoric manifolds M ( P ) correspond-ing to ideal almost Pogorelov polytopes.(3) The family R IP og of canonical 3-dimensional small covers R ( P ) corresponding to idealalmost Pogorelov polytopes.In [DJ91] the manifolds from the second and the third families are called pullbacks from thelinear model .Each family is parametrised by ideal almost Pogorelov polytopes. On the other hand, asmentioned in the introduction, ideal almost Pogorelov polytopes are in bijection with pairs { Q, Q ∗ } , where Q is a combinatorial combinatorial convex (not necessarily simple) 3-polytope,and Q is its dual polytope. Namely for a polytope Q the ideal almost Pogorelov polytope P isobtained by cutting off all the vertices and then cutting off all the ”old” edges. The triangulation ∂P ∗ is the barycentric subdivision of ∂Q ∗ . Since all the arguments in this paper and [E20] workfor any field F taken instead of the ring Z , we obtain the following result. Theorem 8.2. (1)
The family Z IP og of moment-angle manifolds {Z P } corresponding toideal almost Pogorelov polytopes is cohomologically rigid over Z or any field F . (2) The family M IP og of canonical -dimensional quasitoric manifolds { M ( P ) } corre-sponding to ideal almost Pogorelov polytopes is cohomologically rigid over Z or anyfield F . (3) The family R IP og of canonical -dimensional small covers { R ( P ) } corresponding toideal almost Pogorelov polytopes is cohomologically rigid over Z . Hyperbolic geometry of -dimensional manifolds corresponding to idealalmost Pogorelov polytopes For introduction to hyperbolic geometry and Coxeter groups we recommend [VS88].First let us describe explicitly the small cover R ( P ) associated to an ideal almost Pogorelovpolytope P . Remind that R ( P ) = Z × P/ ∼ , where ( t , p ) ∼ ( t , q ) if and only if p = q , and t − t ∈ h Λ i i F i ∋ p . Here Λ i = e c ( i ) , where c ( i ) isthe colour of the face F i . For convenience we will assume that quadrangles are coloured in thethird colour.Thus, R ( P ) is clued from 8 copies of P of the form t × P , t ∈ Z . Each point in t × P lyingin relative interior of • the polytope P belongs only to one polytope t × P ; • the face F i belongs to two polytopes: t × P and ( t + Λ i ) × P ; • the edge F i ∩ F j belongs to four polytopes t × P , ( t + Λ i ) × P , ( t + Λ j ) × P , and( t + Λ i + Λ j ) × P ; • a vertex belongs to all the eight polytopes.Let Q be an ideal right-angled polytope such that P is combinatorially obtained from Q bycutting off vertices at infinity. We can realize P as a part of Q obtained by cutting off verticesat infinity by small horospheres centered at them. The intersection of a horosphere with thepolytope is a Euclidean rectangle.In the polytope P the faces intersecting any quadrangle by opposite edges have the samecolour. When we pass one of these faces F i , we move from t × P to ( t + Λ i ) × P . When we passthe other face F j in the new polytope we move to ( t + Λ i + Λ j ) × P = t × P . Therefore, in R ( P )each quadrangle corresponds to a flat torus T glued from 4 copies of a Eucledean rectangle).Since all the dihedral angles of Q are right, outside this disjoint set of tori, the manifold has ahyperbolic structure. When we shrink the horospheres to points, then the tori are also shrinkedto points, and the manifold R ( P ) is transformed into a space glued from two parts R and R along finite sets of points corresponding to ideal vertices of Q . If we delete these points, theneach part c R i is a hyperbolic manifold of finite volume and is glued of four copies of Q : t × Q ,( t + e ) × Q , ( t + e ) × Q , and ( t + e + e ) × Q . It corresponds to a part of R ( P ) bounded -RIGIDITY OF IDEAL ALMOST POGORELOV POLYTOPES 17 by tori. Passing the quadrangles we add the vector e and move to the other part. Moreover,each part is a manifold with boundary homotopy equivalent to c R i , and R ( P ) is a double ofthis manifold: it is glued from two equal manifolds along boundaries.The hyperbolic manifold c R i of finite volume can be obtained from Q by a construction similarto Construction 0.5 (see [V17, Section 5]). Namely, let Q be an ideal right-angled polytope inthe Lobachevsky space L . There is a right-angled Coxeter group G ( Q ) generated by reflectionsin faces of Q . It it known that G ( Q ) = h ρ , . . . , ρ m i / ( ρ , . . . , ρ m , ρ i ρ j = ρ j ρ i for all F i ∩ F j = ∅ ) , where ρ i is the reflection in the face F i .The colouring of P into three colours induces the colouring of Q into 2 colours (black andwhite).The mapping Λ defines a homomorphism ϕ : G ( Q ) → Z = h e , e i ⊂ Z = h e , e , e i bythe rule ϕ ( ρ i ) = Λ i . It can be shown that Ker ϕ acts on L freely (this follows from the factthat the stabiliser of a point in Q by the action of G ( Q ) is generated by reflections in facescontaining this point). Then L / Ker ϕ is a hyperbolic manifold of finite volume homeomorphicto Z × Q/ ∼ , where ( t , p ) ∼ ( t , q ) if and only if p = q, and t − t ∈ h Λ i i F i ∋ p . A homeomorphism is given by the mapping ( t, p ) → ϕ − ( t ) · p . We see that this manifoldis homeomorphic to c R i . Moreover, the manifolds c R and c R correspond to two embeddings Z → Z : ( x , x ) → ( x , x ,
0) and ( x , x ) → ( x , x , Acknowledgements
The author is grateful to Victor Buchstaber for his encouraging support and attention tothis work, and Taras Panov and Alexander Gaifullin for useful discussions.
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