rigidity problems in toric topologykuroki/gdansk(2012).pdfrigidity problems in toric topology...
TRANSCRIPT
. . . . . .
Rigidity problems in toric topology
Rigidity problems in toric topology
Shintaro Kuroki
OCAMI
September 6-8, 2012Symposium on Geometry of Manifolds and Group Actions
(Gdansk University of Technology)
. . . . . .
Rigidity problems in toric topology
Contents
Table of Contents
1 What is toric topology?
2 Cohomological rigidity problems
. . . . . .
Rigidity problems in toric topology
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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).
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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).
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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).
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
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.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
Example
1 CP2#CP2 is a quasitoric manifold;
2 S2n ⊂ Cn ⊕ R is a torus manifold.
. . . . . .
Rigidity problems in toric topology
What is toric topology?
What is toric topology?
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
. . . . . .
Rigidity problems in toric topology
What is toric topology?
What is toric topology?
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
. . . . . .
Rigidity problems in toric topology
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)
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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?
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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.
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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).
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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).
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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).
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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?
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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).
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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?
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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.
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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?)
. . . . . .
Rigidity problems in toric topology
Cohomological rigidity problems
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.