The Stiefel–Whitney classes of a moment-angle
manifold are trivial
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)
Osaka Prefecture University

Toric Topology 2022 in Osaka, March 23

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Contents§1. Introduction
Definition of a moment-angle manifold
Motivation of research
Goal of this talkideas§2. (Equivariant) Stiefel–Whitney classes for topological manifolds§3. Further problem

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

§1. Introduction
Definition of a moment-angle manifold
Motivation of research
Goal of this talkideas

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Definition of a moment-angle manifold
Hereafter, K and m always denote a simplicial complex and thenumber of its vertices.
For σ ∈ K , put Zσ := {(z1 , . . . , zm ) ∈ (D 2 )m |zi ∈ S 1 if i ̸∈ σ}.
Definition (moment-angle complex)
For a simplicial complex K , the moment-angle complex ZK for K
is defined by∪
ZK :=Zσ .
σ∈KIn this talk, a moment-angle manifold means a moment-angle
complex which is also a topological manifold without boundary,
not necessarily smooth.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Motivation of research
The simplest example of a moment-angle mfd is Z∂∆d = S 2d−1 .
ProblemIs any moment-angle manifold null-cobordant (i.e. the boundary ofsome manifold) ?
This problem is easily proven for smooth ZK (and, as mentionedlater, still open for not smooth cases).
Proof.
Consider the canonical T m = (S 1 )m –action on ZK ⊆ (D 2 )m andlet T ⊆ T m be the diagonal subtorus.
If ZK is smooth, we obtain a principal S 1 –bundle ZK → ZK /T .
This is the sphere bundle for some complex line bundle, andtherefore the total space ZK is the boundary of its disk bundle.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Goal of this talkMoreover, we can prove the following theorem, which does not
require the smoothness.
Theorem (H–Kishimoto–Kizu)
For a (real) moment-angle manifold, the Stiefel–Whiney classes
vanish.
The definition of the Stiefel–Whitney classes for a topologicalmanifold is explained in the next section.
This theorem includes the claim in the previous page, because it
implies the vanishing of Stiefel–Whitney numbers. (Recall that asmooth manifold is null-cobordant iﬀ its Stiefel–Whitney numbersvanish.)
However, this equivalence does not hold for topological manifolds,
so the problem is still open for not smooth ones.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Ideas
The fundamental idea for this research is very simple.
(1) As is well-known, if ZK is smooth, the SW classes of ZK canbe discribed as the images of equivariant ones, under the map
ZK → BT m ZK = (ZK × ET m )/T m .
(2) Let us consider the fiber sequence ZK → BT m ZK → BT m .
It is also known that BT m ZK ≃ DJK , the Davis–Januszkiewicz
space, and BT m ZK → BT m is surjective in cohomology.
By considering the Leray–Serre spectral sequence, we see
ZK → BT m ZK is trivial in cohomology.
In conclusion, the SW classes of ZK are in the image ofH ∗ (BT m ZK ; Z/2) → H ∗ (ZK : Z/2) and this map is trivial, so theSW classes are equal to 0.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Then, to prove the theorem, it suﬃces to show that this argument
also holds for topological moment-angle manifolds, namely, todefine the equivariant SW classes for topological manifolds so thatthey are mapped to the ordinary ones.
(This argument also holds for real moment-angle manifold, but I
will omit the details in order to avoid a repetition.)

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)§2. (Equivariant) Stiefel–Whitney classes for topological manifolds
Stiefel–Whitney classes for topological manifolds
Equivariant Stiefel–Whitney classes for topological manifolds

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Stiefel–Whitney classes for topological manifolds
Let M be an n–dim closed connected topological manifold.
We can define the topological tangent fiber space J of M as
follows so that it is equivalent to the tangent bundle if M issmooth.
J0 := {ℓ : [0, 1] → M | ℓ(t) ̸= ℓ(0) for ∀t ̸= 0}.
J := J0 ∪ {ℓ : [0, 1] → M | ℓ(t) = ℓ(0) for ∀t ̸= 0}.
p : J → M defined by p(ℓ) = ℓ(0).
Theorem (Fadell 1965)
If M is orientable over a ring R, there is Φ ∈ H n (J , J0 ; R) suchthat ϕ : H ∗ (M; R) → H ∗+n (J , J0 ; R) defined by ϕ(u) = p ∗ (u) ∪ Φis an isomorphism.
Then, in the same way for smooth manifolds, we can define thei–th SW class of M by wi (M) = ϕ−1 (Sqi ϕ(1)) .

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

This page is a note on the Thom class.
Theorem (Fadell 1965)
If M is orientable over a ring R, there is Φ ∈ H n (J , J0 ; R) suchthat ϕ : H ∗ (M; R) → H ∗+n (J , J0 ; R) defined by ϕ(u) = p ∗ (u) ∪ Φis isomorphism.
More precisely, the Thom class Φ is characterized by the condition
that, for each x ∈ M, Φ restricts to a generator ofH n (J |x , J0 |x ; R) ∼= R.
Recall that an orientation of a topological manifold M is acollection of generators Ux ∈ H n (M, M \ {x}; R) satisfying certainpasting condition.
Since p restricts to H n (M, M \ {x}; R) ∼= H n (J |x , J0 |x ; R), if wechoose an orientation of M, the Thom class Φ is uniquelydetermined.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Equivariant SW classes for topological manifolds
Suppose that a topological groupG acts on M.
If we define a G –action on J by (g · ℓ)(t) = g · ℓ(t), thenp : J → M is G –equivariant.
Then we obtain the following from G → EG → BG .
M → BG M → BG
(J , J0 ) → (BG J , BG J0 ) → BG
We assume that these fiber sequences are R–orientable, that is,the cohomology local coeﬃcient systems over R are trivial.
The following is immediate from the Leray–Serre spectral sequenceof (J , J0 ) → (BG J , BG J0 ) → BG .
Proposition
There is a unique ΦG ∈ H n (BG J , BG J0 ; R) which restricts to Φ.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Let Er be the Leray–Serre spectral sequence of M → BG M → BGand Êr be that of (J , J0 ) → (BG J , BG J0 ) → BG .
Additionally, let pG : BG J → BG M be the map induced fromp × idEG .
Theorem (H–Kishimoto–Kizu)
ϕG : H ∗ (BG M; R) → H ∗+n (BG J , BG J0 ; R) defined by
ϕG (u) = pG∗ (u) ∪ ΦG is an isomorphism.
To explain the proof shortly, if we take ΨG ∈ C n (BG J , BG J0 ; R)
representing ΦG and define a map
ψG : C ∗ (BG M; R) → C ∗ (BG J , BG J0 ; R) by ψG (u) = pG∗ (u) ∪ ΨG ,then it respects the filtrations of Leray–Serre spectral sequence.
Thus we see that ψG induces a homomorphism E1∗,∗ → Ê1∗,∗+n and
it is an isomorphism in E2 –page by construction.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Now we can define the equivariant SW classes of a topologicalmanifold M in the same way as ordinary ones.
Definitioni
The i–th equivariant SW class of M is wiG (M) = ϕ−1
G (Sq ϕG (1)) .

RemarkAs is almost clear from the construction, wiG (M) 7→ wi (M) under
M → BG M.
Thus we see that the vanishing of SW classes of topologicalmoment-angle manifold can be proven in the same way as smoothones.

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)§3. Further problem

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

Recall that this research starts from the problem whether amoment-angle manifold is null-cobordant or not, and to solve itaﬃrmatively, we use the S 1 –bundle ZK → ZK /T where T is a1–dim subtorus freely acting on ZK .
More generally, let us consider the class of manifolds which aredescribed as ZK /T where T is a subtorus of T m acting on ZK
freely.
For this class, the vanishing of SW classes seems to occur
accidentally. (Probably we can share this intuition if you consider
this problem for quasitoric manifolds.)

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)

On the other hand, the vanishing of SW numbers seems to hold inmany cases, and the following problem seems still worth
considering.
ProblemLet K be an (n − 1)–dim simplicial complex with m vertices andT ⊆ T m act on ZK freely.
If ZK /T is a manifold and dim T < m − n, do the SW numbers of
ZK /T vanish?
If there is T ′ satisfying the same conditions as T and T ⊊ T ′ ,then this problem is easily solved aﬃrmatively since the Borel
construction map ZK /T → BT m /T ZK /T ≃ DJK factors through
ZK /T ′ → BT m /T ′ ZK /T ′ ≃ DJK . (Due to the diﬀerence ofdimension, ZK /T → ZK /T ′ is 0 in the top cohomology group.)

The Stiefel–Whitney classes of a moment-angle manifold are trivi
Sho Hasui(joint work with Daisuke Kishimoto and Akatsuki Kizu)