On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction

On the quasitoric manifolds over a simple
polytope with one vertex cut

Previousstudies
Main part

Sho Hasui
University of Tsukuba

Toric Topology 2019 in Okayama, November 19

On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui

1 Introduction

Introduction
Previousstudies
Main part2 Previous studies

3 Main part

On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main part§1. Introduction

NotationsOn thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies

T n := (S 1 )nThe standard action T n ↷ Cn :(t1 , . . . , tn ) · (z1 , . . . , zn ) := (t1 z1 , . . . , tn zn ).

Main partTn ↷ X,Ydef
f : X → Y is weakly equivariant ⇐⇒∃ψ ∈ Aut(T n ), ∀t ∈ T n , ∀x ∈ X , f (t · x) = ψ(t) · f (x).

Deﬁnition of a quasitoric manifold
On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui

M : 2n–dim smooth manifold
T n ↷ M : smooth
P : simple n–polytope (e.g. ∆n , I n )Introduction
Previousstudies

Deﬁnition (Davis–Januszkiewicz 1991)

Main part

M is a quasitoric manifold over P iflocal(i) [T n ↷ M] ∼= [T n ↷ Cn ] : weakly equivariant diﬀeo,(ii) M/T n ∼= P : homeo as manifolds with corners.
A simple example is CP 2 with the standard T 2 –action.
(following pages)

example: CP 2 (1)On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main partT 2 ↷ CP 2 by (t1 , t2 ) · [z0 : z1 : z2 ] := [z0 : t1 z1 : t2 z2 ]Condition (i)local[T 2 ↷ CP 2 ] ∼= [T 2 ↷ C2 ] : weakly equivariant diﬀeo.
t = (t1 , t2 ) ∈ T 2z = [z0 : z1 : z2 ] ∈ CP 2
Ui = (zi 6= 0), φi : Ui → C2 : i–th standard chart of CP 2
For φ0 : U0 ∼= C2 and z = [1 : z1 : z2 ] ∈ U0 ,φ0 (t · z) = (t1 z1 , t2 z2 ) = t · φ0 (z).

example: CP 2 (2)On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main partFor φ1 : U1 ∼= C2 and z = [z0 : 1 : z2 ] ∈ U1 ,φ1 (t · z) = φ1 ([z0 : t1 : t2 z2 ])= φ1 ([t1−1 z0 : 1 : t1−1 t2 z2 ]) = ψ1 (t) · φ1 (z)where ψ1 (t1 , t2 ) = (t1−1 , t1−1 t2 ), an automorphism of T 2 .

example: CP 2 (2)On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main partFor φ1 : U1 ∼= C2 and z = [z0 : 1 : z2 ] ∈ U1 ,φ1 (t · z) = φ1 ([z0 : t1 : t2 z2 ])= φ1 ([t1−1 z0 : 1 : t1−1 t2 z2 ]) = ψ1 (t) · φ1 (z)where ψ1 (t1 , t2 ) = (t1−1 , t1−1 t2 ), an automorphism of T 2 .
Similarly, for z = [z0 : z1 : 1] ∈ U2 ,φ2 (t · z) = ψ2 (t) · φ2 (z)where ψ2 (t1 , t2 ) = (t2−1 , t1 t2−1 ), an automorphism of T 2 .

example: CP 2 (3)On thequasitoric
manifolds overa simple
polytope withone vertex cut

Condition (ii)
CP 2 /T 2 ∼= ∆2 : homeo as manifolds with corners.

Sho Hasui
Introduction
Previousstudies
Main partCP 2 /T 2 = S 5 /T 3 . (S 5 ⊆ C3 )Then the moment map (z0 , z1 , z2 ) 7→ (|z0 |, |z1 |, |z2 |) descends
to a homeo S 5 /T 3 → S 2 ∩ (R≥0 )3 ∼= ∆2 .

example: CP 2 (3)On thequasitoric
manifolds overa simple
polytope withone vertex cut

Condition (ii)
CP 2 /T 2 ∼= ∆2 : homeo as manifolds with corners.

Sho Hasui
Introduction
Previousstudies
Main partCP 2 /T 2 = S 5 /T 3 . (S 5 ⊆ C3 )Then the moment map (z0 , z1 , z2 ) 7→ (|z0 |, |z1 |, |z2 |) descends
to a homeo S 5 /T 3 → S 2 ∩ (R≥0 )3 ∼= ∆2 .
RemarkCP n is a quasitoric manifold over ∆n .
Fact
Moreover, any projective non-singular toric variety is aquasitoric manifold.

Fundamental correspondence
On thequasitoricmanifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies

Fact
There is the following one-to-one correspondence:{a toric variety}←→{a fan}{a quasitoric manifold} ←→ {a characteristic pair}

Main partA characteristic pair is (P, λ) where P is a simple polytope andλ is a characteristic matrix on P.
A characteristic matrix on P is an integer matrix satisfying acertain condition depending on P, which reﬂects theinformation of isotropy subgroups.
()
1 0 1
2
For example, the characteristic matrix of CP is.
0 1 1

Construction of quasitoric manifold
On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main part

λ: characteristic matrix on P⇝ M(λ) := (P × T n )/ ∼λ , a qt mfd over P with ch mat λ∼λ : isotropy information represented by λThis construction gives the fundamental correspondence in theprevious page.

Motivation
On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main part

Up to weakly equivariant homeo, the classiﬁcation of qt mfd’s
is a purely combinatoric matter through the fundamental
correspondence.
However, the characteristic matrices give very little
informations on the homeo’s which are NOT weakly
equivariant.
Aim
Develop a new method to ﬁnd homeomorphisms between
quasitoric manifolds which are not weakly equivariant.
We denote “not (necessarily) weakly equivariant” by “not eqv.”

On thequasitoric
manifolds overa simple
polytope withone vertex cut
Sho Hasui
IntroductionPreviousstudies
Main part§2. Previous studies

Connected sum (1)On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction

We regard the simple polytopes as a subclass of the manifolds
with corners.
Fact
The connected sum of simple polytopes is a simple polytope.

Previousstudies
Main part

Let M, N be quasitoric manifolds over P, Q.
Then the equivariant connected sum M♯N is a quasitoric
manifold over P♯Q.
On the other hand, we can easily determine whether aquasitoric manifold is decomposed into an equivariant
connected sum of quasitoric manifolds.
(by using the characteristic matrix)

Connected sum (2)On thequasitoricmanifolds overa simple
polytope withone vertex cut
Sho Hasui
Introduction
Previousstudies
Main partWe can get not eqv homeo’s by using the connected sum.
For example, suppose that P = Q♯∆n and there are qt mfd’s
over P corresponding to the ch matrices λ, λ′ .
Moreover, suppose that λ = (A|B|v), λ′ = (A|B|v′ ) where(A|B) is on Q and (B|v), (B|v′ ) are on ∆n .
If det B = ±1,M(λ) ∼= M(A|B)♯M(B|v),

M(λ′ ) ∼= M(A|B)♯M(B|v′ ).

Since M(B|v) ∼= M(B|v′ ) ∼= CP n (the classiﬁcation over ∆n ),′we see that M(λ) ∼= M(λ ).

Toric manifolds over vc(I n )On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui

Let us consider the case P = vc(I n ) = I n ♯∆n .
On the toric manifolds over vc(I n ), we have the following.
(They are a subclass of qt mfds over vc(I n ).)Note that any ch mat can be denoted by (−En |B|v) now.

Introduction
Previousstudies

Theorem (H–Kuwata–Masuda–Park 2018)

Main partdet B is invariant under cohomology isomorphism.
Theorem (H–Kuwata–Masuda–Park 2018)
Let V n (q) denote the set of isomorphism classes of toric
manifolds over vc(I n ) with det B = q.
(1) If q 6= 0, 1, 2, then V n (q) contains only 1 element.
(2) If q = 0, 2, then V n (q) contains only 1 diﬀeo class.

On thequasitoric
manifolds overa simple
polytope withone vertex cut
Sho HasuiIntroduction
Previousstudies
Main part§3. Main part

Setting
On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui
Introduction
Previousstudies
Main partP = vc(Q) = Q♯∆nλ = (A|B|v), λ′ = (A|B|v′ ): characteristic matrices on P
M := M(λ), M ′ = M(λ′ )Consider whether an analogue of the previous theorem holds inthis situation.
Problem
Are M and M ′ homeomorphic?
By an elementary calculation on the characteristic matrices, weobtain the following.
Lemma
If | det B| ≥ 3, then v = v′ .

Idea
On thequasitoric
manifolds overa simple
polytope withone vertex cut

Denote by π : M → P, π ′ : M ′ → P the projections.
Let F be the facet made by vertex cut, N be a tubularneighborhood of F and S := ∂N.

Sho Hasui
Introduction
Previousstudies
Main partThen∪

M = π −1 (P \ N̊)

π −1 (N),π −1 (S)∪M ′ = π ′−1 (P \ N̊)

π ′−1 (N).

π ′−1 (S)From the construction of M(λ) and M(λ′ ), we have
π −1 (P \ N̊) = π ′−1 (P \ N̊),π −1 (S) = π ′−1 (S).

Moreover, π −1 (N) is the disk bundle of a complex line bundle
over π −1 (F ) ∼= CP n−1 classiﬁed by det B.

The case det B 6= 0On thequasitoric
manifolds overa simple
polytope with
one vertex cut

Suppose det B 6= 0 and let G := Z/| det B|.
To compare the attaching diﬀeo, we consider the following
diagram.
9αIntroduction
Previousstudies/ π −1 (F ):π −1 (S)Sho Hasui∼=∼=/ CP n−1S 2n−1 /GMain part∼=β∼=%π ′−1 (S)$/ π ′−1 (F )If det B = ±2, we can take α and β speciﬁcally and show thatβ −1 ◦ α is isotopic to the identity.
Proposition
If det B 6= 0, then M ∼= M ′.

Remark on the case det B = 0
On thequasitoric
manifolds overa simple
polytope with
one vertex cut
Sho Hasui

Recall:P = vc(Q) = Q♯∆nλ = (A|B|v), λ′ = (A|B|v′ ): characteristic matrices on P
M := M(λ), M ′ = M(λ′ )Introduction
Previousstudies
Main partProblem
Are M and M ′ homeomorphic?
For the case det B = 0, we can ﬁnd some counterexample forthis problem in the known classiﬁcation results.
So we have to consider the following in this case.
ProblemFind some additional conditions for λ and λ′ to make M ∼= M′hold.

The case det B = 0 (on-going)
On thequasitoric
manifolds overa simple
polytope with
one vertex cut

In this case, we consider the following diagram.
/ π−1 (F ):π −1 (S)9αSho Hasui
Introduction
Previousstudies
Main part∼=CPn−1×S∼=/ CP1∼=βn−1∼=%

π ′−1 (S)$/ π′−1 (F )Taking speciﬁc α and β, we see β −1 ◦ α(z, t) = (f (t) · z, t)
where f : S 1 → U(n), f (t) = diag(t b1 , . . . , t bn−1 , 1) (bi ∈ Z).
We want to show [f ] = 0 in π1 (U(n)) = Z, which requires
b1 + · · · + bn−1 = 0.
It seems good to consider whether the existence of good
cohomology isomorphism leads to this equality.