Rigidity problems in toric topologykuroki/Gdansk(2012).pdfRigidity problems in toric topology Rigidity problems in toric topology Shintarˆo Kuroki [email protected]

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Rigidity problems in toric topology


Shintaro Kuroki

[email protected]

OCAMI

September 6-8, 2012Symposium on Geometry of Manifolds and Group Actions

(Gdansk University of Technology)

[email protected]

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Contents

Table of Contents

1 What is toric topology?

2 Cohomological rigidity problems

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What is toric topology?

What is toric geometry?

Toric geometry is· · ·

Toric variety (alg. geom.)1:1⇐⇒ Fan (combinatorcs)

Definition (toric variety)

Toric variety is a cpx n-dim (real 2n-dim) normal alg. varietywhich has a (C∗)n(alg. torus)-action with the dence orbit.(C∗ = C\{0} ⊃ T 1).Toric manifold is a non-singular (smooth), toric variety.

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Examples of toric manifolds

1 Cn： by the standard (C∗)n-action;

2 CPn： by the follwing (C∗)n-action:

[z0 : z1 : · · · : zn]g7−→ [z0 : g1z1 : · · · : gnzn];

3 Hirzebruch surface Hk = P(γk ⊕ ϵ)： the projectivizationof the Whitney sum of the tautological line bundle γ withk times tensor product and the trivial (cpx) line bdl ϵ overCP1. This has the natural (C∗)2-action.By definition, Hk is a CP1-bdl over CP1. (e.g.H0 = CP1 × CP1).

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[z0 : z1 : · · · : zn]g7−→ [z0 : g1z1 : · · · : gnzn];


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[z0 : z1 : · · · : zn]g7−→ [z0 : g1z1 : · · · : gnzn];


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From topological point of view

Remark the following two properites,

a toric manfiold has the compact torusT n(⊂ (C∗)n)-action;

M/T n is an n-dim simple polytope (if M is projective),i.e., each vertex can be constructed by an intersection ofexactly n facets.

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Example

The restricted T n-action on CPn = (Cn+1\{0})/C∗ is definedby

[z0 : z1 : · · · : zn]t7−→ [z0 : t1z1 : · · · : tnzn].

Figure: The orbit space CP2/T 2.

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Example

Hk∼= S3 ×S1 P(Ck ⊕ C) has the following T 2-action:

[(z0, z1), (w1,w2)](t1,t2)7−→ [(z0, t1z1), (w1, t2w2)].

Figure: The orbit space Hk/T2.

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Toric manifolds from topological point of view· · ·Definition (1991 Davis-Januszkiewicz)

Qusitoric manifold is the following real 2n-dim. cpt orieT n-manifold:

1 T n-action locally looks like the standard T n-action on Cn;

2 M/T is an n-dim simple covex polytope.

Definition (2003 Hattori-Masuda)

Torus manifold is a real 2n-dim. cpt orie T n-manifold withfixed points.

Definition (2010 Ishida-Fukukawa-Masuda)

Topological toric manifold is a torus mfd which has a smooth(C∗)n-action with the dense orbit.

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Example

1 CP2#CP2 is a quasitoric manifold;

2 S2n ⊂ Cn ⊕ R is a torus manifold.

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Toric topology is· · ·

Topological objects1:1?⇐⇒ Combinatorial objects

Example

Quasitoric mfd1:1⇐⇒ Simple polytope+ch fct

Torus mfd =⇒ Multi-fan or torus graph

Topological toric1:1⇐⇒ Topological fan

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Toric topology is· · ·

Topological objects1:1?⇐⇒ Combinatorial objects

Example

Quasitoric mfd1:1⇐⇒ Simple polytope+ch fct

Torus mfd =⇒ Multi-fan or torus graph

Topological toric1:1⇐⇒ Topological fan

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Cohomological rigidity problems

Cohomological rigidity problem

Motivation· · · Want to find a complete invariant of objectsappeared in toric topology!

Theorem

Let (M ,T ), (M ′,T ) be (quasi)toric manifolds. Then thefollowings are equivalent：

1 (M ,T ) ∼= (M ′,T ) (equiv. homeo);

2 H∗T (M) ≃ H∗

T (M′) (as H∗(BT )-algebra).

Problem (Masuda-Suh)

Let M, M ′ be (quasi)toric manifolds. Then, is it true thatH∗(M) ≃ H∗(M ′) ⇒ M ∼= M ′? （cohomological rigidityproblem）

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Partial affirmative answers· · ·The 2-dimensional (quasi)toric manifold is CP1(≃ S2);

4-dimensional (quasi)toric manifolds are classified byOrlik-Raymond;

For more than 6-dimensional manifolds,

2-stage generalized Bott manifolds (Choi-Masuda-Suh)4-stage Bott manifolds (Choi)

Here, a generalized Bott manifold is

Bnπn−→ Bn−1

πn−1−→ · · · π2−→ B1π1−→ {∗}

where πi : BiCPki−→ Bi−1 is the projectivization of a sum of line

bdls (if each ki = 1, Bn is called Bott manifold).

Problem

How about the other classes of manifolds?

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Rigidity of torus manifolds

Theorem (K)

If a torus manifold M has an extended G-action withcodimension one orbits, then M is diffeomorphic to a CPk orS2k-bundle over

∏S2l ×

∏CPm.

Theorem (Choi-K)

Let M = {M S2k

−→ CPm} be the subset of such manifolds.Then, (roughly)

M ∈ M is cohomological rigid ⇔ k ≤ m;

Topological types of M can be classified by cohomologyrings and real characteristic classes.

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Problem

What is the biggest class which satisfies cohomological rigidityin torus manifolds?

Definition

Let M2nb2=0 be the set of 2n-dim simply connected torus

manifolds such that Hodd(M) = H2(M) = 0.

Then,

M2b2=0 = ∅ (by M2 ∼= S2);

M4b2=0 = {S4} (by Orlik-Raymond);

M6b2=0 = {S6} (by Wall, Jupp).

M8b2=0 = {S8, #ℓ

i=1S4 × S4} (by K).

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Problem


Definition



Then,

M2b2=0 = ∅ (by M2 ∼= S2);



M8b2=0 = {S8, #ℓ

i=1S4 × S4} (by K).

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Problem


Definition



Then,

M2b2=0 = ∅ (by M2 ∼= S2);



M8b2=0 = {S8, #ℓ

i=1S4 × S4} (by K).

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Partial answer· · ·Corollary

M2nb2=0 (n ≤ 4) satisfies cohomological rigidity.

We may ask the following problem

Problem

Let M ∈ M2nb2=0. Then, is the following true

M ∼= #ℓi=1S

2n1i × · · · S2nki i

where∑ki

j=1 nji = n and nji ≥ 2?

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Rigidity of other classes

Definition (K-Suh)

Complex projective (CP)-tower is

Cnπn−→ Cn−1

πn−1−→ · · · π2−→ C1π1−→ {∗}

where πi : CiCPki−→ Ci−1 is the projectivization of a cpx v.b.

Remark

A generalized Bott mfd is a toric mfd, but CP-tower is notalways toric mfd (e.g. Flag manifold, Milnor manifold has thestrucutre of CP-tower but not toric).

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Theorem (K-Suh)

Cohomological rigidity is true for CP-towers up to6-dimension.

Remark (This does not satisfy for 8-dim.)

For example, there are just two cpx 2-dim vector bdls withtrivial Chern classes (by Atiyah-Rees). Let ϵ, η be such bdls.Then, H∗(P(ϵ)) ≃ H∗(P(η)) but P(ϵ) ∼= P(η) (because theirhomotopy groups are different).

Problem

Is (homotopical) rigidity true for CP-tower?

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Toric HK manifold· · · (M4n,T n), i.e., hyperKahler analogue of(symplectic) toric manifolds.

Theorem

(Mα,T , µα) and (Mβ,T , µβ) are toric HK mfd and HKmoment maps. Then, the followings are equivalent

1 (Mα,T , µα) ≡w (Mβ,T , µβ);

2 there is a weak H∗(BT )-alg iso

f ∗T : H∗T (Mα;Z) → H∗

T (Mβ;Z) s.t. (f ∗T )R(a) = b.

Remark

Toric HK mfds do not satisfy cohomological rigidity (e.g. Hn

and Hm). However, if we fix the dimension, then the problemis open.

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Small cover· · · (Mn,Zn2), i.e., real analogue of quaistoric

manifolds.

Theorem (K-Masuda-Yu)

If a fundamental group of aspherical small cover M is virtuallysolvable, then M is a real Bott mfd (an iterated S1-bdl).

Remark

Real Bott mfds satisfy Z2-cohomological rigidity (byKamishima-Masuda). So such class satisfies cohomologicalrigidity. However, small covers are not so. For aspherical smallcovers, the problem is still open. (Small cover version of Borelconjecture?)

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Summary (some open problems)

1 cohomological rigidity problem of (quasi)toric mfds;

2 characterization of the class in torus manifolds whichsatisfy cohomological rigidity;

3 classification of M2nb2=0;

4 rigidity of CP-towers;5 cohomological rigidity of toric HK mfds with fixed

dimension;

6 Z2-cohomological rigidity of aspherical small covers.

Documents

Rigidity problems in toric topologykuroki/Gdansk(2012).pdfRigidity problems in toric topology Rigidity problems in toric topology Shintarˆo Kuroki [email protected]