Cohomological rigidity of manifolds arisen from right-angled 3 - - PowerPoint PPT Presentation

Cohomological rigidity of manifolds arisen from right-angled 3-dimensional polytopes

Seonjeong Park 1 Joint work with Buchstaber 2, Erokhovets 2, Masuda 1, and Panov 2

1Osaka City University, Japan 2Moscow State University, Russia

Topology in Australia and South Korea May 1–5, 2017 The University of Melbourne

Seonjeong Park (OCAMI) Cohomological rigidity May 1 1 / 33

Table of contents

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 2 / 33

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 2 / 33

Introduction

Problem

Given two closed smooth manifolds M and M′, when does an isomorphism H∗(M; Z) ∼ = H∗(M′; Z) imply that M and M′ are diffeomorphic? There are many important series of manifolds for which the cohomology ring does not determine the diffeomorphism class.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 3 / 33

Introduction

Problem

Given two closed smooth manifolds M and M′, when does an isomorphism H∗(M; Z) ∼ = H∗(M′; Z) imply that M and M′ are diffeomorphic? There are many important series of manifolds for which the cohomology ring does not determine the diffeomorphism class. Three-dimensional Lens spaces

L(p; q1) ≃ L(p; q2) ⇔ q1q2 ≡ ±n2 mod p L(p; q1) ∼ = L(p; q2) ⇔ q1 ≡ ±q±1

2

mod p

Seonjeong Park (OCAMI) Cohomological rigidity May 1 3 / 33

Introduction

Problem

Given two closed smooth manifolds M and M′, when does an isomorphism H∗(M; Z) ∼ = H∗(M′; Z) imply that M and M′ are diffeomorphic? There are many important series of manifolds for which the cohomology ring does not determine the diffeomorphism class. Three-dimensional Lens spaces

L(p; q1) ≃ L(p; q2) ⇔ q1q2 ≡ ±n2 mod p L(p; q1) ∼ = L(p; q2) ⇔ q1 ≡ ±q±1

2

mod p

There are infinitely many nonhomeomorphic smooth manifolds with the homotopy type of CP n for n > 2.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 3 / 33

Introduction

Problem

Given two closed smooth manifolds M and M′, when does an isomorphism H∗(M; Z) ∼ = H∗(M′; Z) imply that M and M′ are diffeomorphic? There are many important series of manifolds for which the cohomology ring does not determine the diffeomorphism class. Three-dimensional Lens spaces

L(p; q1) ≃ L(p; q2) ⇔ q1q2 ≡ ±n2 mod p L(p; q1) ∼ = L(p; q2) ⇔ q1 ≡ ±q±1

2

mod p

There are infinitely many nonhomeomorphic smooth manifolds with the homotopy type of CP n for n > 2.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 3 / 33

Introduction

Let γ be the tautological line bundle over CP 1, and let Σn be the total space of the projective bundle P(C ⊕ γ⊗n) for n ∈ Z. Then, Σn is a closed smooth manifold with a smooth effective action of T 2.†

[Hirzebruch, 1951]

The manifolds Σn and Σm are diffeomorphic if and only if n ≡ m (mod 2)

†Σn is called a Hirzebruch surface. Seonjeong Park (OCAMI) Cohomological rigidity May 1 4 / 33

Introduction

Let γ be the tautological line bundle over CP 1, and let Σn be the total space of the projective bundle P(C ⊕ γ⊗n) for n ∈ Z. Then, Σn is a closed smooth manifold with a smooth effective action of T 2.†

[Hirzebruch, 1951]

The manifolds Σn and Σm are diffeomorphic if and only if n ≡ m (mod 2) Note that n ≡ m (mod 2) ⇐ ⇒ H∗(Σn; Z) ∼ = H∗(Σm; Z).

†Σn is called a Hirzebruch surface. Seonjeong Park (OCAMI) Cohomological rigidity May 1 4 / 33

Introduction

Let γ be the tautological line bundle over CP 1, and let Σn be the total space of the projective bundle P(C ⊕ γ⊗n) for n ∈ Z. Then, Σn is a closed smooth manifold with a smooth effective action of T 2.†

[Hirzebruch, 1951]

The manifolds Σn and Σm are diffeomorphic if and only if n ≡ m (mod 2) Note that n ≡ m (mod 2) ⇐ ⇒ H∗(Σn; Z) ∼ = H∗(Σm; Z).

[Petrie, 1973]

For an oriented smooth manifold M of dimR = 6, if H∗(M) ∼ = H∗(CP 3) and M admits a nontrivial smooth semifree circle action, then M is diffeomorphic to CP 3.

†Σn is called a Hirzebruch surface. Seonjeong Park (OCAMI) Cohomological rigidity May 1 4 / 33

Introduction

Let γ be the tautological line bundle over CP 1, and let Σn be the total space of the projective bundle P(C ⊕ γ⊗n) for n ∈ Z. Then, Σn is a closed smooth manifold with a smooth effective action of T 2.†

[Hirzebruch, 1951]

The manifolds Σn and Σm are diffeomorphic if and only if n ≡ m (mod 2) Note that n ≡ m (mod 2) ⇐ ⇒ H∗(Σn; Z) ∼ = H∗(Σm; Z).

[Petrie, 1973]

For an oriented smooth manifold M of dimR = 6, if H∗(M) ∼ = H∗(CP 3) and M admits a nontrivial smooth semifree circle action, then M is diffeomorphic to CP 3.

†Σn is called a Hirzebruch surface. Seonjeong Park (OCAMI) Cohomological rigidity May 1 4 / 33

Introduction

Let k be a commutative ring with unit.†

Definition

A family of closed manifolds is cohomologically rigid over k if manifolds in the family are distinguished up to homeomorphism by their cohomology rings with coefficients in k.

Goal of this talk

We establish cohomological rigidity for particular two families of manifolds

• f dim 3 and 6 arising from the Pogorelov class P consisting of the

polytopes which have right-angled realizations in Lobachevsky space L3.

†If k is not specified explicitly, we assume k = Z. Seonjeong Park (OCAMI) Cohomological rigidity May 1 5 / 33

Introduction

Let k be a commutative ring with unit.†

Definition

A family of closed manifolds is cohomologically rigid over k if manifolds in the family are distinguished up to homeomorphism by their cohomology rings with coefficients in k.

Goal of this talk

We establish cohomological rigidity for particular two families of manifolds

• f dim 3 and 6 arising from the Pogorelov class P consisting of the

polytopes which have right-angled realizations in Lobachevsky space L3.

†If k is not specified explicitly, we assume k = Z. Seonjeong Park (OCAMI) Cohomological rigidity May 1 5 / 33

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 5 / 33

Simple polytopes

Definition

A polytope is a convex hull of finite points in Rn. e.g.) A polygon is a 2-dimensional polytope. Platonic solids are 3-dimensional polytopes.

Definition

An n-polytope is simple if every vertex is the intersection of precisely n facets, codimension-1 faces. simple not simple

Seonjeong Park (OCAMI) Cohomological rigidity May 1 6 / 33

Simple polytopes

Definition

A polytope is a convex hull of finite points in Rn. e.g.) A polygon is a 2-dimensional polytope. Platonic solids are 3-dimensional polytopes.

Definition

An n-polytope is simple if every vertex is the intersection of precisely n facets, codimension-1 faces. simple not simple

Seonjeong Park (OCAMI) Cohomological rigidity May 1 6 / 33

k-belts

For k ≥ 3, a k-belt in a simple 3-polytope is the set of facets k facets, Bk, such that the union of all the facets in Bk is homotopy equivalent to S1 and any union of k − 1 facets in Bk is contractible. There are one 3-belt and three 4-belts. There is neither 3-belt nor 4-belt.

‡

‡The image of dodecahedron is from Wikipedia. Seonjeong Park (OCAMI) Cohomological rigidity May 1 7 / 33

Pogorelov class

Definition

The Pogorelov class P consists of simple 3-polytopes P = ∆3 without 3-and 4-belts. This class includes mathematical fullerene, i.e. simple 3-polytopes with only pentagonal or hexagonal facets. (e.g., dodecahedron, truncated icosahedron) The number of combinatorially different fullerenes with p6 hexagonal facets grows as p9

• 6. [Thurston, 1998]

Seonjeong Park (OCAMI) Cohomological rigidity May 1 8 / 33

Pogorelov class

Definition

The Pogorelov class P consists of simple 3-polytopes P = ∆3 without 3-and 4-belts. This class includes mathematical fullerene, i.e. simple 3-polytopes with only pentagonal or hexagonal facets. (e.g., dodecahedron, truncated icosahedron) The number of combinatorially different fullerenes with p6 hexagonal facets grows as p9

• 6. [Thurston, 1998]

For any finite sequence of nonnegative integers pk, k ≥ 7, there exists a Pogorelov polytope whose number of k-gonal facets is pk. [BEMPP]

Seonjeong Park (OCAMI) Cohomological rigidity May 1 8 / 33

Pogorelov class

Definition

The Pogorelov class P consists of simple 3-polytopes P = ∆3 without 3-and 4-belts. This class includes mathematical fullerene, i.e. simple 3-polytopes with only pentagonal or hexagonal facets. (e.g., dodecahedron, truncated icosahedron) The number of combinatorially different fullerenes with p6 hexagonal facets grows as p9

• 6. [Thurston, 1998]

For any finite sequence of nonnegative integers pk, k ≥ 7, there exists a Pogorelov polytope whose number of k-gonal facets is pk. [BEMPP]

Seonjeong Park (OCAMI) Cohomological rigidity May 1 8 / 33

Right-angled 3-dimensional polytopes

[Pogorelov 1967, Andreev 1970]

A combinatorial 3-polytope can be realized as a right-angled polytope in Lobachevsky space L3 if and only if it is a simple polytope without 3-and 4-belts. Furthermore, such a realization is unique up to isometry.

§

§http://bulatov.org/math/1101/webtalk.html Seonjeong Park (OCAMI) Cohomological rigidity May 1 9 / 33

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 9 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi. Consider G(P) = g1, . . . , gm | g2

i = 1, gigj = gjgi if Fi ∩ Fj = ∅.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi. Consider G(P) = g1, . . . , gm | g2

i = 1, gigj = gjgi if Fi ∩ Fj = ∅.

For an epimorphism ϕ(k) : G(P) → Zk

2 for some k, if at each vertex

v = Fi ∩ Fj ∩ Fℓ, the images of gi, gj, gℓ are linearly independent,

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi. Consider G(P) = g1, . . . , gm | g2

i = 1, gigj = gjgi if Fi ∩ Fj = ∅.

For an epimorphism ϕ(k) : G(P) → Zk

2 for some k, if at each vertex

v = Fi ∩ Fj ∩ Fℓ, the images of gi, gj, gℓ are linearly independent, then ker ϕ(k) acts freely on L3.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi. Consider G(P) = g1, . . . , gm | g2

i = 1, gigj = gjgi if Fi ∩ Fj = ∅.

For an epimorphism ϕ(k) : G(P) → Zk

2 for some k, if at each vertex

v = Fi ∩ Fj ∩ Fℓ, the images of gi, gj, gℓ are linearly independent, then ker ϕ(k) acts freely on L3. Then the quotient N = L3/ ker ϕ(k) is a closed hyperbolic 3-manifold which is composed of 2k copies

• f P.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi. Consider G(P) = g1, . . . , gm | g2

i = 1, gigj = gjgi if Fi ∩ Fj = ∅.

For an epimorphism ϕ(k) : G(P) → Zk

2 for some k, if at each vertex

v = Fi ∩ Fj ∩ Fℓ, the images of gi, gj, gℓ are linearly independent, then ker ϕ(k) acts freely on L3. Then the quotient N = L3/ ker ϕ(k) is a closed hyperbolic 3-manifold which is composed of 2k copies

• f P. Note that N is aspherical.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Hyperbolic 3-manifolds

1 In 1931, L¨

• bell glued together the sides of eight copies of a

right-angled 3-polytope in L3 to create the first example of a compact, orientable, hyperbolic 3-manifold.

2 [Vesnin, 1987] Let P be a right-angled 3-polytope in L3 with facets

F1, . . . , Fm. Denote by gi the reflection in the facet Fi. Consider G(P) = g1, . . . , gm | g2

i = 1, gigj = gjgi if Fi ∩ Fj = ∅.

For an epimorphism ϕ(k) : G(P) → Zk

2 for some k, if at each vertex

v = Fi ∩ Fj ∩ Fℓ, the images of gi, gj, gℓ are linearly independent, then ker ϕ(k) acts freely on L3. Then the quotient N = L3/ ker ϕ(k) is a closed hyperbolic 3-manifold which is composed of 2k copies

• f P. Note that N is aspherical.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 10 / 33

Smooth manifolds arising from a simple n-polytope

[Davis-Januszkiewicz, 1991] d Rd Fd Gd 1 Z2 R Z2 2 Z C S1 Let P be a simple n-polytope with facets F1, . . . , Fm. Consider a function λd : {F1, . . . , Fm} → Rn

d satisfying

(∗)

• Fi : vertex =

⇒ {λ(Fi)} : a basis of Rn

d.

For each face F = ∩Fi, let GF be the subgroup of Gn

d determined by the

span of λd(Fi)’s.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 11 / 33

Smooth manifolds arising from a simple n-polytope

[Davis-Januszkiewicz, 1991] d Rd Fd Gd 1 Z2 R Z2 2 Z C S1 Let P be a simple n-polytope with facets F1, . . . , Fm. Consider a function λd : {F1, . . . , Fm} → Rn

d satisfying

(∗)

• Fi : vertex =

⇒ {λ(Fi)} : a basis of Rn

d.

For each face F = ∩Fi, let GF be the subgroup of Gn

d determined by the

span of λd(Fi)’s. Then from the equivalence relation on P × Gn

d by

(x, g) ∼ (x′, g′) ⇔ x = x′ and g−1g′ ∈ GF(x), we get a smooth dn-dimensional manifold Mdn(P, λd) = P × Gn

d/ ∼,

where F(x) is the face of P containing x in its relative interior.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 11 / 33

Smooth manifolds arising from a simple n-polytope

[Davis-Januszkiewicz, 1991] d Rd Fd Gd 1 Z2 R Z2 2 Z C S1 Let P be a simple n-polytope with facets F1, . . . , Fm. Consider a function λd : {F1, . . . , Fm} → Rn

d satisfying

(∗)

• Fi : vertex =

⇒ {λ(Fi)} : a basis of Rn

d.

For each face F = ∩Fi, let GF be the subgroup of Gn

d determined by the

span of λd(Fi)’s. Then from the equivalence relation on P × Gn

d by

(x, g) ∼ (x′, g′) ⇔ x = x′ and g−1g′ ∈ GF(x), we get a smooth dn-dimensional manifold Mdn(P, λd) = P × Gn

d/ ∼,

where F(x) is the face of P containing x in its relative interior.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 11 / 33

Examples

A hyperbolic 3-manifold of L¨

• bell type is appeared when d = 1 and P

is a right-angled 3-polytope. Every projective smooth toric variety is appeared when d = 2 and P is a Delzant polyope†; e.g. Hirzebruch surface Σa is

F2 F3 F4 F1

• −1
• 1

a

• 1
• −1
• The manifold CP 2#CP 2 is appeared when d = 2, P = , and

λ2(F1) = −1

• , λ2(F2) =

−1

• , λ2(F3) =

1 2

• , λ2(F4) =

1 1

• †An n-dimensional convex polytope is said to be a Delzant polytope if the (outward)

normal vectors to the facets meeting at each vertex form an integral basis of Zn.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 12 / 33

Small covers and Quasitoric manifolds

[Davis-Januszkiewicz, 1991]

A closed smooth manifold M of dimension 2n (resp. n) is called a quasitoric manifold M (resp. small cover) if it has a smooth action of T n (resp. Zn

2) such that

1 the action of T n (resp. Zn

2) is locally standard, and

2 there is a projection π: M → P such that the fibers of π are the

T n-orbits (resp. Zn

2-orbits),

where P is a simple polytope of dimension n. d Rd Fd Gd 1 Z2 R Z2 2 Z C S1 The natural action of Gn

d on Fn d is called the standard n-dimensional

representation.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 13 / 33

Quasitoric manifolds and small covers

For a quasitoric manifold M = M2n(P, λ2), there is an involution τ on M whose fixed point set is the small cover Mn(P, λ1), where λ1 is the mod 2 reduction of λ2. For the Hirzebruch surface Σa, if a is even (resp. odd), then the corresponding small cover is the torus T 2 (resp. the Klein bottle).

Seonjeong Park (OCAMI) Cohomological rigidity May 1 14 / 33

Equivalent quasitoric manifolds

Two quasitoric manifolds M and M′ over P are equivalent if there exist a homeomorphism f : M → M′ and an automorphism θ of T n such that f(g · x) = θ(g) · f(x) for every x ∈ M and every g ∈ T n and f covers the identity on P.

Theorem [Davis-Januszkiewicz]

A quasitoric manifold is determined up to equivalence over P by its characteristic function λ.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 15 / 33

Equivalent quasitoric manifolds

Two quasitoric manifolds M and M′ over P are equivalent if there exist a homeomorphism f : M → M′ and an automorphism θ of T n such that f(g · x) = θ(g) · f(x) for every x ∈ M and every g ∈ T n and f covers the identity on P.

Theorem [Davis-Januszkiewicz]

A quasitoric manifold is determined up to equivalence over P by its characteristic function λ. Let Λ be the matrix whose ith column is λ(Fi). Then M(P, λ) and M(P ′, λ′) are equivalent if and only if

1 there is a combinatorial equivalence P ≈ P ′ preserving the ordering of

facets, and

2 Λ′ = AΛB, where A ∈ GLn(Z) and B is an m × m diagonal matrix

with ±1 on the diagonal.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 15 / 33

Equivalent quasitoric manifolds

Two quasitoric manifolds M and M′ over P are equivalent if there exist a homeomorphism f : M → M′ and an automorphism θ of T n such that f(g · x) = θ(g) · f(x) for every x ∈ M and every g ∈ T n and f covers the identity on P.

Theorem [Davis-Januszkiewicz]

A quasitoric manifold is determined up to equivalence over P by its characteristic function λ. Let Λ be the matrix whose ith column is λ(Fi). Then M(P, λ) and M(P ′, λ′) are equivalent if and only if

1 there is a combinatorial equivalence P ≈ P ′ preserving the ordering of

facets, and

2 Λ′ = AΛB, where A ∈ GLn(Z) and B is an m × m diagonal matrix

with ±1 on the diagonal.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 15 / 33

Cohomology of quasitoric manifolds

The cohomology ring of a quasitoric manifold M = M(P, λ) is H∗(M(P, λ)) ∼ = Z[v1, . . . , vm]/IP + Jλ, deg(vi) = 2, where IP = vi1 · · · vik | Fi1 ∩ · · · ∩ Fik = ∅ in P and Jλ = m

• i=1

λi, xvi

• x ∈ Zn
• .

F2 F3 F4 F1

• −1
• 1

1

• 1
• −1
• H∗(M(P, λ))

∼ = Z[v1, . . . , v4]/v1v3, v2v4, v1 − v3, v2 − v3 − v4 ∼ = Z[v3, v4]/v2

3, v4(v3 + v4)

Seonjeong Park (OCAMI) Cohomological rigidity May 1 16 / 33

Cohomology of quasitoric manifolds

The cohomology ring of a quasitoric manifold M = M(P, λ) is H∗(M(P, λ)) ∼ = Z[v1, . . . , vm]/IP + Jλ, deg(vi) = 2, where IP = vi1 · · · vik | Fi1 ∩ · · · ∩ Fik = ∅ in P and Jλ = m

• i=1

λi, xvi

• x ∈ Zn
• .

F2 F3 F4 F1

• −1
• 1

1

• 1
• −1
• H∗(M(P, λ))

∼ = Z[v1, . . . , v4]/v1v3, v2v4, v1 − v3, v2 − v3 − v4 ∼ = Z[v3, v4]/v2

3, v4(v3 + v4)

Seonjeong Park (OCAMI) Cohomological rigidity May 1 16 / 33

Moment-angle manifold

Let P be a simple n-polytope with facets F1, . . . , Fm. Let Ti be the coordinate circle subgroup of T m corresponding to Fi. Then for each face F = ∩jFj = ∅ of P, we set TF =

j Tj.

Definition

The moment-angle manifold corresponding to P is ZP = P × T m/ ∼, where (x, t) ∼ (x′, t′) ⇔ x = x′ & t−1t′ ∈ TF(x). Here F(x) is the face containing x in its interior.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 17 / 33

Moment-angle manifold

Let P be a simple n-polytope with facets F1, . . . , Fm. Let Ti be the coordinate circle subgroup of T m corresponding to Fi. Then for each face F = ∩jFj = ∅ of P, we set TF =

j Tj.

Definition

The moment-angle manifold corresponding to P is ZP = P × T m/ ∼, where (x, t) ∼ (x′, t′) ⇔ x = x′ & t−1t′ ∈ TF(x). Here F(x) is the face containing x in its interior.

Example

Z∆n = S2n+1 and Zk

i=1 ∆ni =

k

• i=1

S2ni+1.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 17 / 33

Moment-angle manifold

Let P be a simple n-polytope with facets F1, . . . , Fm. Let Ti be the coordinate circle subgroup of T m corresponding to Fi. Then for each face F = ∩jFj = ∅ of P, we set TF =

j Tj.

Definition

The moment-angle manifold corresponding to P is ZP = P × T m/ ∼, where (x, t) ∼ (x′, t′) ⇔ x = x′ & t−1t′ ∈ TF(x). Here F(x) is the face containing x in its interior.

Example

Z∆n = S2n+1 and Zk

i=1 ∆ni =

k

• i=1

S2ni+1.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 17 / 33

Relationship between M(P, λ) and ZP

The matrix Λ =

• λ1

· · · λm

• corresponding to λ induces a surjective

homomorphism λ : T m → T n. = ⇒ ker(λ) is an (m − n)-dimensional subtorus of T m.

Theorem [Davis-Januszkiewicz]

The subtorus ker(λ) acts freely on ZP , thereby defining a principal T m−n-bundle ZP → M(P, λ).

Seonjeong Park (OCAMI) Cohomological rigidity May 1 18 / 33

Relationship between M(P, λ) and ZP

The matrix Λ =

• λ1

· · · λm

• corresponding to λ induces a surjective

homomorphism λ : T m → T n. = ⇒ ker(λ) is an (m − n)-dimensional subtorus of T m.

Theorem [Davis-Januszkiewicz]

The subtorus ker(λ) acts freely on ZP , thereby defining a principal T m−n-bundle ZP → M(P, λ). The following matrix defines a characteristic function on the standard simplex ∆n Λ =      1 · · · −1 1 · · · −1 . . . . . . ... . . . . . . · · · 1 −1     

n×(n+1)

. Then ker(λ) = {(t, t, . . . , t)} ⊂ T n+1 and S2n+1/ ker(λ) = CP n.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 18 / 33

Relationship between M(P, λ) and ZP

The matrix Λ =

• λ1

· · · λm

• corresponding to λ induces a surjective

homomorphism λ : T m → T n. = ⇒ ker(λ) is an (m − n)-dimensional subtorus of T m.

Theorem [Davis-Januszkiewicz]

The subtorus ker(λ) acts freely on ZP , thereby defining a principal T m−n-bundle ZP → M(P, λ). The following matrix defines a characteristic function on the standard simplex ∆n Λ =      1 · · · −1 1 · · · −1 . . . . . . ... . . . . . . · · · 1 −1     

n×(n+1)

. Then ker(λ) = {(t, t, . . . , t)} ⊂ T n+1 and S2n+1/ ker(λ) = CP n.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 18 / 33

Cohomology of moment-angle manifolds

Recall that H∗(M(P, λ)) = Z[v1, . . . , vm]/IP + Jλ. Let k[P] = k[v1, . . . , vm]/IP .

Seonjeong Park (OCAMI) Cohomological rigidity May 1 19 / 33

Cohomology of moment-angle manifolds

Recall that H∗(M(P, λ)) = Z[v1, . . . , vm]/IP + Jλ. Let k[P] = k[v1, . . . , vm]/IP .

Theorem [Buchstaber-Panov]

1 There are isomorphisms of (multi)graded commutative algebras

H∗(ZP ) ∼ = Tork[v1,...,vm](k[P], k) ∼ = H[Λ[u1, . . . , um] ⊗ k[P], d), where mdeg(ui) = (−1, 2ei), mdeg(vi) = (0, 2ei), dui = vi, dvi = 0.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 19 / 33

Cohomology of moment-angle manifolds

Recall that H∗(M(P, λ)) = Z[v1, . . . , vm]/IP + Jλ. Let k[P] = k[v1, . . . , vm]/IP .

Theorem [Buchstaber-Panov]

1 There are isomorphisms of (multi)graded commutative algebras

H∗(ZP ) ∼ = Tork[v1,...,vm](k[P], k) ∼ = H[Λ[u1, . . . , um] ⊗ k[P], d), where mdeg(ui) = (−1, 2ei), mdeg(vi) = (0, 2ei), dui = vi, dvi = 0.

2 There is an isomorphism of cohomology rings

H∗(ZP ; Z) ∼ = TorZ[v1,...,vm]/Jλ(Z[P]/Jλ, Z).

Seonjeong Park (OCAMI) Cohomological rigidity May 1 19 / 33

Cohomology of moment-angle manifolds

Recall that H∗(M(P, λ)) = Z[v1, . . . , vm]/IP + Jλ. Let k[P] = k[v1, . . . , vm]/IP .

Theorem [Buchstaber-Panov]

1 There are isomorphisms of (multi)graded commutative algebras

H∗(ZP ) ∼ = Tork[v1,...,vm](k[P], k) ∼ = H[Λ[u1, . . . , um] ⊗ k[P], d), where mdeg(ui) = (−1, 2ei), mdeg(vi) = (0, 2ei), dui = vi, dvi = 0.

2 There is an isomorphism of cohomology rings

H∗(ZP ; Z) ∼ = TorZ[v1,...,vm]/Jλ(Z[P]/Jλ, Z).

Seonjeong Park (OCAMI) Cohomological rigidity May 1 19 / 33

Summary

Given a simple polytope P, there is a moment-angle manifold ZP of dimension n + m; if P admits a characteristic function λd, then

there is a quasitoric manifold M(P, λ2) of dimension 2n, there is a small cover M(P, λ1)of dimension n, and λ1 is the mod 2 reduction of λ2.

The quasitoric manifold M(P, λ) is ZP / ker(λ2).

Goal of this talk

We establish cohomological rigidity for particular two families of manifolds

• f dim 3 and 6 arising from the Pogorelov class P consisting of the

polytopes which have right-angled realizations in Lobachevsky space L3.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 20 / 33

Summary

Given a simple polytope P, there is a moment-angle manifold ZP of dimension n + m; if P admits a characteristic function λd, then

there is a quasitoric manifold M(P, λ2) of dimension 2n, there is a small cover M(P, λ1)of dimension n, and λ1 is the mod 2 reduction of λ2.

The quasitoric manifold M(P, λ) is ZP / ker(λ2).

Goal of this talk

We establish cohomological rigidity for particular two families of manifolds

• f dim 3 and 6 arising from the Pogorelov class P consisting of the

polytopes which have right-angled realizations in Lobachevsky space L3.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 20 / 33

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 20 / 33

Rigidity problems

In 2006, Masuda and Suh introduced the following problem.

Cohomological rigidity problems for quasitoric manifolds

If two quasitoric manifolds M and M′ have the same cohomology ring with integral coefficients, are they homeomorphic? In other words, is the family of quasitoric manifolds cohomologically rigid? This problem is still OPEN. There is no counter example, but there are many results which support the affirmative answer.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 21 / 33

Rigidity problems

In 2006, Masuda and Suh introduced the following problem.

Cohomological rigidity problems for quasitoric manifolds

If two quasitoric manifolds M and M′ have the same cohomology ring with integral coefficients, are they homeomorphic? In other words, is the family of quasitoric manifolds cohomologically rigid? This problem is still OPEN. There is no counter example, but there are many results which support the affirmative answer.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 21 / 33

Known results

1 Quasitoric manifolds of dimR ≤ 4 [Orlik-Raymond (1970)] 2 m

i=1 CP ni [Masuda-Panov (2008), Choi-Masuda-Suh (2010)]

3 Projective smooth toric varieties with second Betti number 2

[Choi-Masuda-Suh (2010)]

4 Quasitoric manifolds with second Betti number 2 [Choi-P-Suh (2012)] 5 Quasitoric manifolds over the cube I3 and dual cyclic polytopes

[Hasui (2015)]

6 Projective bundles over smooth compact toric surfaces [Choi-P

(2016)] . . .

Seonjeong Park (OCAMI) Cohomological rigidity May 1 22 / 33

Rigidity problems

Cohomological rigidity problems for moment-angle manifolds

Let ZP1 and ZP2 be two moment-angle manifolds whose (bigraded) cohomology rings are isomorphic. Are they homeomorphic? In other words, is the family of moment-angle manifolds cohomologically rigid? This problem is also open.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 23 / 33

Rigidity problems

Cohomological rigidity problems for moment-angle manifolds

Let ZP1 and ZP2 be two moment-angle manifolds whose (bigraded) cohomology rings are isomorphic. Are they homeomorphic? In other words, is the family of moment-angle manifolds cohomologically rigid? This problem is also open.

Cohomological rigidity for small covers

Two small covers N and N′ over the n-cube In are diffeomorphic if and only if H∗(N; Z2) ∼ = H∗(N′; Z2). [Kamishima-Masuda, 2009] There are many pair of small covers N and N′ over the product of simplices such that H∗(N; Z2) ∼ = H∗(N′; Z2) but N and N′ are not homotopy equivalent. [Masuda, 2010]

Seonjeong Park (OCAMI) Cohomological rigidity May 1 23 / 33

Rigidity problems

Cohomological rigidity problems for moment-angle manifolds

Let ZP1 and ZP2 be two moment-angle manifolds whose (bigraded) cohomology rings are isomorphic. Are they homeomorphic? In other words, is the family of moment-angle manifolds cohomologically rigid? This problem is also open.

Cohomological rigidity for small covers

Two small covers N and N′ over the n-cube In are diffeomorphic if and only if H∗(N; Z2) ∼ = H∗(N′; Z2). [Kamishima-Masuda, 2009] There are many pair of small covers N and N′ over the product of simplices such that H∗(N; Z2) ∼ = H∗(N′; Z2) but N and N′ are not homotopy equivalent. [Masuda, 2010]

Seonjeong Park (OCAMI) Cohomological rigidity May 1 23 / 33

Example

Note that M(P, λ) ∼ = M(P ′, λ′) or ZP1 ∼ = ZP2 does not imply that the polytopes P1 and P2 are combinatorially equivalent. The orbit space of CP 3#3CP 3 is a three times vertex-cut of ∆3. The corresponding moment-angle manifolds are homeomorphic to the connected sum of sphere products #6

k=3(Sk × S9−k)#(k−2)(

5 k−1). Seonjeong Park (OCAMI) Cohomological rigidity May 1 24 / 33

Example

Note that M(P, λ) ∼ = M(P ′, λ′) or ZP1 ∼ = ZP2 does not imply that the polytopes P1 and P2 are combinatorially equivalent. The orbit space of CP 3#3CP 3 is a three times vertex-cut of ∆3. The corresponding moment-angle manifolds are homeomorphic to the connected sum of sphere products #6

k=3(Sk × S9−k)#(k−2)(

5 k−1). Seonjeong Park (OCAMI) Cohomological rigidity May 1 24 / 33

Rigidity problems for polytopes

Definition [Masuda-Suh]

A simple polytope P is said to be C-rigid if it satisfies any of the following there is no quasitoric manifold whose orbit space is P; or there exists a quasitoric manifold whose orbit space is P, and whenever there exists a quasitoric manifold M′ over another polytope P ′ with a graded ring isomorphism H∗(M) ∼ = H∗(M′), there is combinatorial equivalence P ≈ P ′.

Definition [Buchstaber]

A simple polytope P is said to be B-rigid if any cohomology ring isomorphism H∗(ZP ) ∼ = H∗(ZQ) of moment-angle manifolds implies a combinatorial equivalence P ≈ Q.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 25 / 33

Rigidity problems for polytopes

Definition [Masuda-Suh]

A simple polytope P is said to be C-rigid if it satisfies any of the following there is no quasitoric manifold whose orbit space is P; or there exists a quasitoric manifold whose orbit space is P, and whenever there exists a quasitoric manifold M′ over another polytope P ′ with a graded ring isomorphism H∗(M) ∼ = H∗(M′), there is combinatorial equivalence P ≈ P ′.

Definition [Buchstaber]

A simple polytope P is said to be B-rigid if any cohomology ring isomorphism H∗(ZP ) ∼ = H∗(ZQ) of moment-angle manifolds implies a combinatorial equivalence P ≈ Q.

Proposition [Choi-Panov-Suh]

If a simple polytope P is B-rigid, then it is C-rigid.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 25 / 33

Rigidity problems for polytopes

Definition [Masuda-Suh]

A simple polytope P is said to be C-rigid if it satisfies any of the following there is no quasitoric manifold whose orbit space is P; or there exists a quasitoric manifold whose orbit space is P, and whenever there exists a quasitoric manifold M′ over another polytope P ′ with a graded ring isomorphism H∗(M) ∼ = H∗(M′), there is combinatorial equivalence P ≈ P ′.

Definition [Buchstaber]

A simple polytope P is said to be B-rigid if any cohomology ring isomorphism H∗(ZP ) ∼ = H∗(ZQ) of moment-angle manifolds implies a combinatorial equivalence P ≈ Q.

Proposition [Choi-Panov-Suh]

If a simple polytope P is B-rigid, then it is C-rigid.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 25 / 33

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 25 / 33

Over the Pogorelov class

Theorem [Fan-Ma-Wang]

The polytopes in P are B-rigid. Note that Every simple polytope of dimension 3 admits a characteristic function by the Four Color Theorem. (e1, e2, e3, 3

i=1 ei)

Seonjeong Park (OCAMI) Cohomological rigidity May 1 26 / 33

Over the Pogorelov class

Theorem [Fan-Ma-Wang]

The polytopes in P are B-rigid. Note that Every simple polytope of dimension 3 admits a characteristic function by the Four Color Theorem. (e1, e2, e3, 3

i=1 ei)

There are smooth toric varieties whose orbit spaces are in P. [Suyama]

Seonjeong Park (OCAMI) Cohomological rigidity May 1 26 / 33

Over the Pogorelov class

Theorem [Fan-Ma-Wang]

The polytopes in P are B-rigid. Note that Every simple polytope of dimension 3 admits a characteristic function by the Four Color Theorem. (e1, e2, e3, 3

i=1 ei)

There are smooth toric varieties whose orbit spaces are in P. [Suyama] We can consider the families of quasitoric manifolds whose orbit spaces are polytopes in the class P.

Corollary

The polytopes in P are C-rigid.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 26 / 33

Over the Pogorelov class

Theorem [Fan-Ma-Wang]

The polytopes in P are B-rigid. Note that Every simple polytope of dimension 3 admits a characteristic function by the Four Color Theorem. (e1, e2, e3, 3

i=1 ei)

There are smooth toric varieties whose orbit spaces are in P. [Suyama] We can consider the families of quasitoric manifolds whose orbit spaces are polytopes in the class P.

Corollary

The polytopes in P are C-rigid.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 26 / 33

Cohomological rigidity

Lemma [Fan-Ma-Wang]

Consider the cohomology classes T (P) = {±[uivj] ∈ H3(ZP ) | Fi ∩ Fj = ∅}. If P ∈ P, then for any cohomology ring isomorphism ψ: H∗(ZP ) → H∗(ZP ′), we have ψ(T (P)) = T (P ′).

Lemma

Consider the set of cohomology classes D(M) = {±[vi] ∈ H2(M) | i = 1, . . . , m}. If P ∈ P and M′ is a quasitoric manifold over P ′, then for any cohomology ring isomorphism ϕ: H∗(M) → H∗(M′) we have ϕ(D(M)) = D(M′). Moreover, the set D(M) is mapped bijectively under ϕ.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 27 / 33

Cohomological rigidity

Lemma [Fan-Ma-Wang]

Consider the cohomology classes T (P) = {±[uivj] ∈ H3(ZP ) | Fi ∩ Fj = ∅}. If P ∈ P, then for any cohomology ring isomorphism ψ: H∗(ZP ) → H∗(ZP ′), we have ψ(T (P)) = T (P ′).

Lemma

Consider the set of cohomology classes D(M) = {±[vi] ∈ H2(M) | i = 1, . . . , m}. If P ∈ P and M′ is a quasitoric manifold over P ′, then for any cohomology ring isomorphism ϕ: H∗(M) → H∗(M′) we have ϕ(D(M)) = D(M′). Moreover, the set D(M) is mapped bijectively under ϕ.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 27 / 33

Cohomological rigidity

Theorem

Let M = M(P, λ) and M′ = M(P ′, λ′). Assume that P belongs to the Pogorelov class P. Then the following are equivalent.

1 H∗(M) and H∗(M′) are isomorphic; 2 M and M′ are diffeomorphic; and 3 M and M′ are equivalent.

Remark

Let Σn and Σm be Hirzebruch surfaces. Then Σn and Σm are diffeomorphic if and only if n ≡ m mod 2, and Σn and Σm are equivalent if and only if |n| = |m|.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 28 / 33

Cohomological rigidity

Theorem

Let M = M(P, λ) and M′ = M(P ′, λ′). Assume that P belongs to the Pogorelov class P. Then the following are equivalent.

1 H∗(M) and H∗(M′) are isomorphic; 2 M and M′ are diffeomorphic; and 3 M and M′ are equivalent.

Remark

Let Σn and Σm be Hirzebruch surfaces. Then Σn and Σm are diffeomorphic if and only if n ≡ m mod 2, and Σn and Σm are equivalent if and only if |n| = |m|.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 28 / 33

Cohomological rigidity for small covers

Theorem

Let N and N′ be small covers of P and P ′, respectively. Assume that P belongs to the Pogorelov class P. Then the following are equivalent.

1 H∗(N; Z2) and H∗(N′; Z2) are isomorphic; 2 π1(N) and π1(N′) are isomorphic; 3 N and N′ are diffeomorphic; and 4 N and N′ are equivalent. Seonjeong Park (OCAMI) Cohomological rigidity May 1 29 / 33

Vesnin’s conjecture

For k ≥ 5, let Qk be a simple 3-polytope with top and bottom k-gonal facets and 2k pentagonal facets forming two k-belts around the top and bottom.

Vesnin’s conjecture

The manifolds N(Qk, χ) and N(Q′

k, χ′) are isometric if and only if the

4-colorings χ and χ′ are equivalent.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 30 / 33

Vesnin’s conjecture

Vesnin (1987), Magulis (1974), Antonlin-Camarena, Maloney, Gregory, Roland (2009) proved the Vesnin’s conjecture except for k = 6, 8.

Corollary

The hyperbolic manifold N(Qk, χ) and N(Qk, χ′) defined by regualr 4-colorings of the polytope Qk, k ≥ 5, are isometric if and only if the 4-colorings χ and χ′ are equivalent.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 31 / 33

1 Introduction 2 Right-angled 3-polytopes 3 Smooth manifolds arising from simple polytopes 4 Cohomological rigidity problems 5 Over the Pogorelov class 6 Remarks

Seonjeong Park (OCAMI) Cohomological rigidity May 1 31 / 33

Cohomological rigidity for 6-dim’l quasitoric manifolds

Theorem [Wall, Jupp]

Let ϕ: H∗(N) → H∗(N′) be an isomorphism of the cohomology rings of smooth closed simply connected 6-dimensional manifolds N and N′ with H3(N) = H3(N′) = 0. Suppose that

1 ϕ(w2(N)) = w2(N′), where w2(N) ∈ H2(N; Z2) is the second

Stiefel-Whiteny class; and

2 ϕ(p1(N)) = ϕ(p1(N′)), where p1(N) ∈ H4(N) is the first Pontryagin

class. Then the manifolds N and N′ are diffeomorphic.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 32 / 33

Cohomological rigidity for 6-dim’l quasitoric manifolds

Lemma [Choi-Masuda-Suh]

Suppose that the ring H∗(N; Z2) is generated by Hk(N; Z2) for some k > 0. Then any cohomology ring isomorphism ϕ: H∗(N; Z2) → H∗(N′; Z2)) preserves the total Stiefel-Whitney class, i.e., ϕ(w(N)) = w(N′).

Corollary

The family of 6-dimensional quasitoric manifolds is cohomologically rigid if any cohomology ring isomorphism between them preserves the first Pontryagin class.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 33 / 33

Cohomological rigidity for 6-dim’l quasitoric manifolds

Lemma [Choi-Masuda-Suh]

Suppose that the ring H∗(N; Z2) is generated by Hk(N; Z2) for some k > 0. Then any cohomology ring isomorphism ϕ: H∗(N; Z2) → H∗(N′; Z2)) preserves the total Stiefel-Whitney class, i.e., ϕ(w(N)) = w(N′).

Corollary

The family of 6-dimensional quasitoric manifolds is cohomologically rigid if any cohomology ring isomorphism between them preserves the first Pontryagin class.

Seonjeong Park (OCAMI) Cohomological rigidity May 1 33 / 33

Thank you very much!