Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Topics about complex and real toric spaces

Zhang, Song
Fudan University
Toric Topology 2022 in Osaka
March 23, 2022

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

1 / 31

Topics aboutcomplex and real
toric spaces

Contents

Zhang, SongIntroduction

1

Introduction
Moment-angle complexes and polyhedral products
Complex and real toric spaces over simple polytopes

2

Filtrations of complex and real toric spaces
The cubical decomposition of Pn
A filtration of a quotient of moment-angle manifold
A cellular decomposition of a real toric space3

The cohomology of quotients of moment-angle manifolds over polygons
The induced spectral sequence
The cohomology of moment-angle manifolds over polygons(Co)homology groups of the quotients of moment-angle manifold under special colourings overpolygons

4

Fundamental groups of real toric spaces over simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

2 / 31

Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Moment-angle complexes and polyhedral products

Moment-angle complexes are firstly formally formulated by Buchstaber and Panov in 1998:

Introduction
Moment-angle
complexes andpolyhedral products

Definition of a moment-angle complex

ZK = Z(K; (D2 , S1 )) =Complex and real
toric spaces over
simple polytopes

[Y

D2i ×

Y

I∈K i∈I

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Si1 ⊂ (D2 )mi∈I/

where K is a simplicial complex on vertices set [m] = {1, . . . , m}.
A generalized functorial construction is a polyhedral product (aka. a generalized moment-angle complex),
which has been studied since 1966 in G. Porter’s work:
Definition of a polyhedral productFundamentalgroups of real
toric spaces over
simple polytopes

Z(K; (X, A)) = (X, A)K =

[Y

Xi ×

I∈K i∈I

Y

Ai

i∈I
/

where (X, A) = {(Xi , Ai ) : i = 1, . . . , m} is a family of based CW-pairs.
Examples

ZK = (D2 )m when K = ∆m−1
ZK = S2m−1 when K = ∂∆m−1
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

3 / 31

Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Moment-angle complexes and polyhedral products

Moment-angle complexes are firstly formally formulated by Buchstaber and Panov in 1998:

Introduction
Moment-angle
complexes andpolyhedral products

Definition of a moment-angle complex

ZK = Z(K; (D2 , S1 )) =Complex and real
toric spaces over
simple polytopes

[Y

D2i ×

Y

I∈K i∈I

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Si1 ⊂ (D2 )mi∈I/

where K is a simplicial complex on vertices set [m] = {1, . . . , m}.
A generalized functorial construction is a polyhedral product (aka. a generalized moment-angle complex),
which has been studied since 1966 in G. Porter’s work:
Definition of a polyhedral productFundamentalgroups of real
toric spaces over
simple polytopes

Z(K; (X, A)) = (X, A)K =

[Y

Xi ×

I∈K i∈I

Y

Ai

i∈I
/

where (X, A) = {(Xi , Ai ) : i = 1, . . . , m} is a family of based CW-pairs.
Examples

ZK = (D2 )m when K = ∆m−1
ZK = S2m−1 when K = ∂∆m−1
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

3 / 31

Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Moment-angle complexes and polyhedral products

Moment-angle complexes are firstly formally formulated by Buchstaber and Panov in 1998:

Introduction
Moment-angle
complexes andpolyhedral products

Definition of a moment-angle complex

ZK = Z(K; (D2 , S1 )) =Complex and real
toric spaces over
simple polytopes

[Y

D2i ×

Y

I∈K i∈I

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Si1 ⊂ (D2 )mi∈I/

where K is a simplicial complex on vertices set [m] = {1, . . . , m}.
A generalized functorial construction is a polyhedral product (aka. a generalized moment-angle complex),
which has been studied since 1966 in G. Porter’s work:
Definition of a polyhedral productFundamentalgroups of real
toric spaces over
simple polytopes

Z(K; (X, A)) = (X, A)K =

[Y

Xi ×

I∈K i∈I

Y

Ai

i∈I
/

where (X, A) = {(Xi , Ai ) : i = 1, . . . , m} is a family of based CW-pairs.
Examples

ZK = (D2 )m when K = ∆m−1
ZK = S2m−1 when K = ∂∆m−1
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

3 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Cohomology of moment-angle complexes

Zhang, SongIntroduction

In what way the cohomology of moment-angle complexes perform?

Moment-angle
complexes andpolyhedral products

Definition of Stanley-Reisner algebra (face ring) of K

Complex and real
toric spaces over
simple polytopes

X
Z[K] = Z[v1 , · · · , vm ]/IK = Z[v1 , · · · , vm ]/Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes{i1 ,··· ,is }∈K/(vi1 · · · vis )The Tor-algebra: by the Koszul resolution of Z[K], and 1-to-1 correspondence between the cells of ZK and
the generators of the quotient of the Koszul complex[Baskakov, Buchstaber, and Panov, 2004]

H ∗ (ZK ; Z) ∼= TorZ[v1 ,··· ,vm ] (Z[K], Z)
Furthermore, H ∗ (ZK ) can be graded by the Hochster formula[Buchstaber and Panov, 2002, or in their book Toric Topology]

H p (ZK ) ∼
=

M
I⊂[m]

e p−|I|−1 (KI )H

where KI is the full subcomplex of K on I.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

4 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Cohomology of moment-angle complexes

Zhang, SongIntroduction

In what way the cohomology of moment-angle complexes perform?

Moment-angle
complexes andpolyhedral products

Definition of Stanley-Reisner algebra (face ring) of K

Complex and real
toric spaces over
simple polytopes

X
Z[K] = Z[v1 , · · · , vm ]/IK = Z[v1 , · · · , vm ]/Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes{i1 ,··· ,is }∈K/(vi1 · · · vis )The Tor-algebra: by the Koszul resolution of Z[K], and 1-to-1 correspondence between the cells of ZK and
the generators of the quotient of the Koszul complex[Baskakov, Buchstaber, and Panov, 2004]

H ∗ (ZK ; Z) ∼= TorZ[v1 ,··· ,vm ] (Z[K], Z)
Furthermore, H ∗ (ZK ) can be graded by the Hochster formula[Buchstaber and Panov, 2002, or in their book Toric Topology]

H p (ZK ) ∼
=

M
I⊂[m]

e p−|I|−1 (KI )H

where KI is the full subcomplex of K on I.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

4 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Cohomology of moment-angle complexes

Zhang, SongIntroduction

In what way the cohomology of moment-angle complexes perform?

Moment-angle
complexes andpolyhedral products

Definition of Stanley-Reisner algebra (face ring) of K

Complex and real
toric spaces over
simple polytopes

X
Z[K] = Z[v1 , · · · , vm ]/IK = Z[v1 , · · · , vm ]/Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes{i1 ,··· ,is }∈K/(vi1 · · · vis )The Tor-algebra: by the Koszul resolution of Z[K], and 1-to-1 correspondence between the cells of ZK and
the generators of the quotient of the Koszul complex[Baskakov, Buchstaber, and Panov, 2004]

H ∗ (ZK ; Z) ∼= TorZ[v1 ,··· ,vm ] (Z[K], Z)
Furthermore, H ∗ (ZK ) can be graded by the Hochster formula[Buchstaber and Panov, 2002, or in their book Toric Topology]

H p (ZK ) ∼
=

M
I⊂[m]

e p−|I|−1 (KI )H

where KI is the full subcomplex of K on I.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

4 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Cohomology of moment-angle complexes

Zhang, SongIntroduction

In what way the cohomology of moment-angle complexes perform?

Moment-angle
complexes andpolyhedral products

Definition of Stanley-Reisner algebra (face ring) of K

Complex and real
toric spaces over
simple polytopes

X
Z[K] = Z[v1 , · · · , vm ]/IK = Z[v1 , · · · , vm ]/Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes{i1 ,··· ,is }∈K/(vi1 · · · vis )The Tor-algebra: by the Koszul resolution of Z[K], and 1-to-1 correspondence between the cells of ZK and
the generators of the quotient of the Koszul complex[Baskakov, Buchstaber, and Panov, 2004]

H ∗ (ZK ; Z) ∼= TorZ[v1 ,··· ,vm ] (Z[K], Z)
Furthermore, H ∗ (ZK ) can be graded by the Hochster formula[Buchstaber and Panov, 2002, or in their book Toric Topology]

H p (ZK ) ∼
=

M
I⊂[m]

e p−|I|−1 (KI )H

where KI is the full subcomplex of K on I.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

4 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Generalizations of ZK = Z(K; (D2 , S1 ))Zhang, SongIntroduction(D,S)

Moment-angle
complexes andpolyhedral products

Let ZK= Z(K; (D, S)), where (D, S) = {(Di , Si )}m
i=1 satisfy the following:(

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopese j (Di ) = 0 for any j,He κ (Si ) ∼e j (Si ) = 0 for any j 6= κi .
there exists a unique integer κi such that H= Z and H
i[X. Cao and Z. Lü, 2012](D,S)

H ∗ (ZK; k) ∼= Tork[v1 ,··· ,vm ] (k[K], k),with the Hochster formula(D,S)

H n (ZK; k) ∼=Tor−i,a(k[K], k) ∼=k[v ,··· ,v ]

M
m

1

M

m

a ∈ {0, 1}
X(κl + 1) = n
l∈a

−1 +

e |a|−i−1 (Ka ; k)Hma ∈ {0, 1}
X(κl + 1) = n
l∈a

−1 +

where k is a field with arbitrary characteristic.

Zhang, Song (Fudan University)

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5 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Generalizations of ZK = Z(K; (D2 , S1 ))Zhang, SongIntroduction(D,S)

Moment-angle
complexes andpolyhedral products

Let ZK= Z(K; (D, S)), where (D, S) = {(Di , Si )}m
i=1 satisfy the following:(

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopese j (Di ) = 0 for any j,He κ (Si ) ∼e j (Si ) = 0 for any j 6= κi .
there exists a unique integer κi such that H= Z and H
i[X. Cao and Z. Lü, 2012](D,S)

H ∗ (ZK; k) ∼= Tork[v1 ,··· ,vm ] (k[K], k),with the Hochster formula(D,S)

H n (ZK; k) ∼=Tor−i,a(k[K], k) ∼=k[v ,··· ,v ]

M
m

1

M

m

a ∈ {0, 1}
X(κl + 1) = n
l∈a

−1 +

e |a|−i−1 (Ka ; k)Hma ∈ {0, 1}
X(κl + 1) = n
l∈a

−1 +

where k is a field with arbitrary characteristic.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

5 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Generalizations of ZK = Z(K; (D2 , S1 ))Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

For a moment-angle complex over the simplicial poset S, given by
[ Y
Y
ZS =
D2j ×
Sj1 ⊂ (D2 )mσ∈S jσ

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopesjσand the face ring of S is given by [Stanley, 1991]
Z[S] = Z[vσ |σ ∈ S]/IS = Z[vσ |σ ∈ S]/ (v0̂ − 1) +X(vσ vτ − vσ∧τ ·σ,τ ∈SXvη ) .

η∈σ∨τ

Still the Tor-algebra and the Hochster formula are available:[Z. Lü and T. Panov, 2011]

H ∗ (ZS ; Z) ∼= TorZ[v1 ,··· ,vm ] (Z[S], Z)
with the Hochster formulaH p (ZS ; Z) ∼=M

Tor−i,2a(Z[S], Z) ∼=Z[v ,··· ,v ]1

−i+2|a|=pZhang, Song (Fudan University)

Mm

Topics about complex and real toric spaces

e |a|−i−1 (|Sa |)H−i+2|a|=pToric Topology 2022 in Osaka, March 23, 2022

6 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Generalizations of ZK = Z(K; (D2 , S1 ))Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

For a moment-angle complex over the simplicial poset S, given by
[ Y
Y
ZS =
D2j ×
Sj1 ⊂ (D2 )mσ∈S jσ

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopesjσand the face ring of S is given by [Stanley, 1991]
Z[S] = Z[vσ |σ ∈ S]/IS = Z[vσ |σ ∈ S]/ (v0̂ − 1) +X(vσ vτ − vσ∧τ ·σ,τ ∈SXvη ) .

η∈σ∨τ

Still the Tor-algebra and the Hochster formula are available:[Z. Lü and T. Panov, 2011]

H ∗ (ZS ; Z) ∼= TorZ[v1 ,··· ,vm ] (Z[S], Z)
with the Hochster formulaH p (ZS ; Z) ∼=M

Tor−i,2a(Z[S], Z) ∼=Z[v ,··· ,v ]1

−i+2|a|=pZhang, Song (Fudan University)

Mm

Topics about complex and real toric spaces

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Topics aboutcomplex and real
toric spaces

Introduction
Generalizations of ZK = Z(K; (D2 , S1 ))Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

For the general case, there is a stable decomposition of a polyhedral product:[Bahri, Bendersky, Cohen, and Gitler, 2010]

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Σ(Z(K; (X, A))) ' Σ _b I ; (X, A)I )
Z(K

I⊂[m]

where Σ means the reduced suspension, Zb means the smash polyhedral product.
which will lead to the decomposition of its cohomology:[Bahri, Bendersky, Cohen, and Gitler, 2012]

H ∗ (Z(K; (X, A))) ∼=Mb I ; (X, A)I ))H ∗ (Z(KI⊂[m]Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

Introduction
Generalizations of ZK = Z(K; (D2 , S1 ))Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

For the general case, there is a stable decomposition of a polyhedral product:[Bahri, Bendersky, Cohen, and Gitler, 2010]

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Σ(Z(K; (X, A))) ' Σ _b I ; (X, A)I )
Z(K

I⊂[m]

where Σ means the reduced suspension, Zb means the smash polyhedral product.
which will lead to the decomposition of its cohomology:[Bahri, Bendersky, Cohen, and Gitler, 2012]

H ∗ (Z(K; (X, A))) ∼=Mb I ; (X, A)I ))H ∗ (Z(KI⊂[m]Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

7 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Moment-angle manifolds over simple polytopes

Zhang, SongDefinition: moment-angle manifolds over Pn with m facets
Introduction
Moment-angle
complexes andpolyhedral products

ZP = Z(KP ; (D2 , S1 )) ⊂ (D2 )mwith a canonical action (S1 )m × ZP → ZP , byComplex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes(ei2πt , x) = (ei2πt1 , · · · , ei2πtm , x1 , · · · , xm ) 7→ (ei2πt1 · x1 , · · · , ei2πtm · xm )where KP is the boundary complex of the dual of Pn .
An equivalent way to construct ZP by the trivial colouring on Pn [Toric Topology, section 6.2]

(S1 )m × Pn / ∼(S1 )m -equivariantly≈

ZP

constructed by a characteristic function λ : F (P) → Zm , F

i 7→ ei , and(0p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )(ei2πt , p) ∼ (ei2πt , p0 ) ⇔∀i ∈/ {i1 , . . . , il }, ti0 − ti ≡ 0 (mod Z)

An example: when P1 = [0, 1], KP consists of only 2 vertices

ZP = D2 × S1 ∪ S1 × D2 ≈ S3 , and (S1 )2 × P1 / ∼ is constructed by colouring (1, 0) on 0 and (0, 1) on 1.
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

8 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Moment-angle manifolds over simple polytopes

Zhang, SongDefinition: moment-angle manifolds over Pn with m facets
Introduction
Moment-angle
complexes andpolyhedral products

ZP = Z(KP ; (D2 , S1 )) ⊂ (D2 )mwith a canonical action (S1 )m × ZP → ZP , byComplex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes(ei2πt , x) = (ei2πt1 , · · · , ei2πtm , x1 , · · · , xm ) 7→ (ei2πt1 · x1 , · · · , ei2πtm · xm )where KP is the boundary complex of the dual of Pn .
An equivalent way to construct ZP by the trivial colouring on Pn [Toric Topology, section 6.2]

(S1 )m × Pn / ∼(S1 )m -equivariantly≈

ZP

constructed by a characteristic function λ : F (P) → Zm , F

i 7→ ei , and(0p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )(ei2πt , p) ∼ (ei2πt , p0 ) ⇔∀i ∈/ {i1 , . . . , il }, ti0 − ti ≡ 0 (mod Z)

An example: when P1 = [0, 1], KP consists of only 2 vertices

ZP = D2 × S1 ∪ S1 × D2 ≈ S3 , and (S1 )2 × P1 / ∼ is constructed by colouring (1, 0) on 0 and (0, 1) on 1.
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

8 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Moment-angle manifolds over simple polytopes

Zhang, SongDefinition: moment-angle manifolds over Pn with m facets
Introduction
Moment-angle
complexes andpolyhedral products

ZP = Z(KP ; (D2 , S1 )) ⊂ (D2 )mwith a canonical action (S1 )m × ZP → ZP , byComplex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes(ei2πt , x) = (ei2πt1 , · · · , ei2πtm , x1 , · · · , xm ) 7→ (ei2πt1 · x1 , · · · , ei2πtm · xm )where KP is the boundary complex of the dual of Pn .
An equivalent way to construct ZP by the trivial colouring on Pn [Toric Topology, section 6.2]

(S1 )m × Pn / ∼(S1 )m -equivariantly≈

ZP

constructed by a characteristic function λ : F (P) → Zm , F

i 7→ ei , and(0p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )(ei2πt , p) ∼ (ei2πt , p0 ) ⇔∀i ∈/ {i1 , . . . , il }, ti0 − ti ≡ 0 (mod Z)

An example: when P1 = [0, 1], KP consists of only 2 vertices

ZP = D2 × S1 ∪ S1 × D2 ≈ S3 , and (S1 )2 × P1 / ∼ is constructed by colouring (1, 0) on 0 and (0, 1) on 1.
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

Introduction
Moment-angle manifolds over simple polytopes

Zhang, Song

What about the real case?
Introduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

Definition: the real version

A real moment-angle complex: RZK = Z(K; (D1 , S0 )) =

S Q
I∈K

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

1i∈I Di ×

Q

0
1 m
i∈I
/ Si ⊂ (D ) , and a real

moment-angle manifold over Pn : RZP = Z(KP ; (D1 , S0 )) ⊂ (D1 )m , with the canonical group action(Z/2)m × RZP → RZP given by(· · · , 0i , · · · , 1j , · · · ), (· · · , xi , · · · , xj , · · · ) 7→ (· · · , xi , · · · , −xj , · · · )An equivalent way to construct RZP [Toric Topology, section 6.2]

Fundamentalgroups of real
toric spaces over
simple polytopes(Z/2)m -equivariantly(Z/2)m × Pn / ∼(where (g, p) ∼ (g0 , p0 ) ⇔≈RZP

p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )g0 − g ∈ Span{ei1 , . . . , eil }An example: when P1 = [0, 1], KP consists of only 2 vertices

RZP = D1 × S0 ∪ S0 × D1 ≈ S1 , and (Z/2)2 × P1 / ∼ is constructed by colouring (1, 0) on 0 and (0, 1) on 1.
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

Introduction
Moment-angle manifolds over simple polytopes

Zhang, Song

What about the real case?
Introduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

Definition: the real version

A real moment-angle complex: RZK = Z(K; (D1 , S0 )) =

S Q
I∈K

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

1i∈I Di ×

Q

0
1 m
i∈I
/ Si ⊂ (D ) , and a real

moment-angle manifold over Pn : RZP = Z(KP ; (D1 , S0 )) ⊂ (D1 )m , with the canonical group action(Z/2)m × RZP → RZP given by(· · · , 0i , · · · , 1j , · · · ), (· · · , xi , · · · , xj , · · · ) 7→ (· · · , xi , · · · , −xj , · · · )An equivalent way to construct RZP [Toric Topology, section 6.2]

Fundamentalgroups of real
toric spaces over
simple polytopes(Z/2)m -equivariantly(Z/2)m × Pn / ∼(where (g, p) ∼ (g0 , p0 ) ⇔≈RZP

p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )g0 − g ∈ Span{ei1 , . . . , eil }An example: when P1 = [0, 1], KP consists of only 2 vertices

RZP = D1 × S0 ∪ S0 × D1 ≈ S1 , and (Z/2)2 × P1 / ∼ is constructed by colouring (1, 0) on 0 and (0, 1) on 1.
Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Introduction
Moment-angle manifolds over simple polytopes

Zhang, Song

What about the real case?
Introduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

Definition: the real version

A real moment-angle complex: RZK = Z(K; (D1 , S0 )) =

S Q
I∈K

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

1i∈I Di ×

Q

0
1 m
i∈I
/ Si ⊂ (D ) , and a real

moment-angle manifold over Pn : RZP = Z(KP ; (D1 , S0 )) ⊂ (D1 )m , with the canonical group action(Z/2)m × RZP → RZP given by(· · · , 0i , · · · , 1j , · · · ), (· · · , xi , · · · , xj , · · · ) 7→ (· · · , xi , · · · , −xj , · · · )An equivalent way to construct RZP [Toric Topology, section 6.2]

Fundamentalgroups of real
toric spaces over
simple polytopes(Z/2)m -equivariantly(Z/2)m × Pn / ∼(where (g, p) ∼ (g0 , p0 ) ⇔≈RZP

p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )g0 − g ∈ Span{ei1 , . . . , eil }An example: when P1 = [0, 1], KP consists of only 2 vertices

RZP = D1 × S0 ∪ S0 × D1 ≈ S1 , and (Z/2)2 × P1 / ∼ is constructed by colouring (1, 0) on 0 and (0, 1) on 1.
Zhang, Song (Fudan University)

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9 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Quotients of moment-angle manifolds

Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products

Definition: Quotients of moment-angle manifolds, introduced by Panov in 2015

A k-subtorus T H freely acts on ZP will associate with a quotient moment-angle (m + n − k)-manifold ZP /T H ,where H = (h1 , · · · , hk ) is an m × k integral matrix, andComplex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

T H = {ei2π(φ1 h1 +···+φk hk ) : φ1 , . . . , φk ∈ R} ⊂ (S1 )m .
The property which H must satisfy is given in Toric Topology (Lemma 4.8.4):
T H acts on ZP freely, if and only if for any simplex {i1 , . . . , il } ∈ KP , vectors ei1 , . . . , eil , h1 , . . . , hk are
unimodular in Zm .
By the Eilenberg–Moore spectral sequence of the homotopy fibration ZK /T H → BT K → B(T m /T H ),[Panov, 2015]

H ∗ (ZK /T H ; R) ∼= TorH ∗ (B(T m /T H );R) (R(K), R)

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Toric Topology 2022 in Osaka, March 23, 2022

10 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Quotients of moment-angle manifolds

Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products

Definition: Quotients of moment-angle manifolds, introduced by Panov in 2015

A k-subtorus T H freely acts on ZP will associate with a quotient moment-angle (m + n − k)-manifold ZP /T H ,where H = (h1 , · · · , hk ) is an m × k integral matrix, andComplex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

T H = {ei2π(φ1 h1 +···+φk hk ) : φ1 , . . . , φk ∈ R} ⊂ (S1 )m .
The property which H must satisfy is given in Toric Topology (Lemma 4.8.4):
T H acts on ZP freely, if and only if for any simplex {i1 , . . . , il } ∈ KP , vectors ei1 , . . . , eil , h1 , . . . , hk are
unimodular in Zm .
By the Eilenberg–Moore spectral sequence of the homotopy fibration ZK /T H → BT K → B(T m /T H ),[Panov, 2015]

H ∗ (ZK /T H ; R) ∼= TorH ∗ (B(T m /T H );R) (R(K), R)

Zhang, Song (Fudan University)

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Toric Topology 2022 in Osaka, March 23, 2022

10 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Quotients of moment-angle manifolds

Zhang, SongIntroduction

Any other simpler way of construction?

Moment-angle
complexes andpolyhedral products

A non-degenerate colouring on Pn

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces

A homomorphism Λ : Zm → Zm−k satisfies that for every vertex Fi1 ∩ · · · ∩ Fin , Span{Λ(ei1 ), · · · , Λ(ein )} is
an n-subspace which is a direct summand of Zm .

The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Each Λ will determine an (n + m − k)-manifold M(Pn , Λ) = (S1 )m−k × Pn / ∼Λ , where(0p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )(ei2πt , p) ∼ (ei2πt , p0 ) ⇔0ei2πt · e−i2πt ∈ T (Λ(ei1 ),··· ,Λ(eil ))Fundamentalgroups of real
toric spaces over
simple polytopes

An equivalent way of construction

Each m × k integral matrix H such that T H acts freely on ZP uniquely determines a non-degenerate colouringΛ : Zm → Zm−k which is an epimorphism, and vice versa. Furthermore, under this situation,
M(Pn , Λ)Zhang, Song (Fudan University)

1 m
H(S1 )m−k ∼=(S ) /T -equivariantly≈

Topics about complex and real toric spaces

ZP /T H

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Topics aboutcomplex and real
toric spaces

Introduction
Quotients of moment-angle manifolds

Zhang, SongIntroduction

Any other simpler way of construction?

Moment-angle
complexes andpolyhedral products

A non-degenerate colouring on Pn

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces

A homomorphism Λ : Zm → Zm−k satisfies that for every vertex Fi1 ∩ · · · ∩ Fin , Span{Λ(ei1 ), · · · , Λ(ein )} is
an n-subspace which is a direct summand of Zm .

The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Each Λ will determine an (n + m − k)-manifold M(Pn , Λ) = (S1 )m−k × Pn / ∼Λ , where(0p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )(ei2πt , p) ∼ (ei2πt , p0 ) ⇔0ei2πt · e−i2πt ∈ T (Λ(ei1 ),··· ,Λ(eil ))Fundamentalgroups of real
toric spaces over
simple polytopes

An equivalent way of construction

Each m × k integral matrix H such that T H acts freely on ZP uniquely determines a non-degenerate colouringΛ : Zm → Zm−k which is an epimorphism, and vice versa. Furthermore, under this situation,
M(Pn , Λ)Zhang, Song (Fudan University)

1 m
H(S1 )m−k ∼=(S ) /T -equivariantly≈

Topics about complex and real toric spaces

ZP /T H

Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

Introduction
Quotients of moment-angle manifolds

Zhang, SongIntroduction

Any other simpler way of construction?

Moment-angle
complexes andpolyhedral products

A non-degenerate colouring on Pn

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces

A homomorphism Λ : Zm → Zm−k satisfies that for every vertex Fi1 ∩ · · · ∩ Fin , Span{Λ(ei1 ), · · · , Λ(ein )} is
an n-subspace which is a direct summand of Zm .

The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Each Λ will determine an (n + m − k)-manifold M(Pn , Λ) = (S1 )m−k × Pn / ∼Λ , where(0p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )(ei2πt , p) ∼ (ei2πt , p0 ) ⇔0ei2πt · e−i2πt ∈ T (Λ(ei1 ),··· ,Λ(eil ))Fundamentalgroups of real
toric spaces over
simple polytopes

An equivalent way of construction

Each m × k integral matrix H such that T H acts freely on ZP uniquely determines a non-degenerate colouringΛ : Zm → Zm−k which is an epimorphism, and vice versa. Furthermore, under this situation,
M(Pn , Λ)Zhang, Song (Fudan University)

1 m
H(S1 )m−k ∼=(S ) /T -equivariantly≈

Topics about complex and real toric spaces

ZP /T H

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Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Quasi-toric manifolds

Quasi-toric manifolds M(Pn , Λ): introduced by Davis and Januszkiewicz in 1991, a special class of quotientsof moment-angle manifolds, constucted by Λ : Zm → Zn .

Introduction
Moment-angle
complexes andpolyhedral products[Davis and Januszkiewicz, 1991]

Complex and real
toric spaces over
simple polytopesH ∗ (M(Pn , Λ); Z) ∼= Z[v1 , · · · , vm ]Filtrations ofcomplex and real
toric spaces

where J =



I+J

Pn

i=1 (Λi1 v1 + · · · + Λim vm ).

The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

Introduction
Real toric spaces over simple polytopes

Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

What about the real case?
Definition: a real toric space over Pn

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Each λ : (Z/2)m → (Z/2)m−k which satisfies the condition
for every vertex Fi1 ∩ · · · ∩ Fin , λ(Fi1 ), . . . , λ(Fin ) are linearly independent in (Z/2)m−k
will associate with a real toric space Mλ = (Z/2)m−k × Pn / ∼λ , where(p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )0 0(g, p) ∼ (g , p ) ⇔g0 − g ∈ Span{λ(Fi1 ), . . . , λ(Fil )}Similarly to the complex case,A rank-k-subgroup H ⊂ (Z/2)m which acts freely on RZP will give rise to an n-manifold RZP /H.

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Topics aboutcomplex and real
toric spaces

Introduction
Real toric spaces over simple polytopes

Zhang, SongIntroduction
Moment-angle
complexes andpolyhedral products
Complex and real
toric spaces over
simple polytopes

What about the real case?
Definition: a real toric space over Pn

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Each λ : (Z/2)m → (Z/2)m−k which satisfies the condition
for every vertex Fi1 ∩ · · · ∩ Fin , λ(Fi1 ), . . . , λ(Fin ) are linearly independent in (Z/2)m−k
will associate with a real toric space Mλ = (Z/2)m−k × Pn / ∼λ , where(p = p0 ∈ Int(Fi1 ∩ · · · ∩ Fil )0 0(g, p) ∼ (g , p ) ⇔g0 − g ∈ Span{λ(Fi1 ), . . . , λ(Fil )}Similarly to the complex case,A rank-k-subgroup H ⊂ (Z/2)m which acts freely on RZP will give rise to an n-manifold RZP /H.

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Topics aboutcomplex and real
toric spaces

Introduction
Real toric spaces over simple polytopes

Zhang, SongAn equivalent way of construction
Introduction
Moment-angle
complexes andpolyhedral products

Each k-subspace H of (Z/2)m which acts freely on RZP uniquely determines a non-degenerate colouring
λ : (Z/2)m → (Z/2)m−k which is an epimorphism, and vice versa. Furthermore, under this situation,

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Mλm(Z/2)m−k ∼=(Z/2) /H-equivariantly≈

RZP /H

Small covers: Introduced by Davis and Januszkiewicz in 1991, a special class of real toric spaces, for they
can be constructed by non-degenerate (Z/2)n -colourings, or equivalently, orbit spaces of RZP by free(Z/2)m−n -actions.

Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

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Toric Topology 2022 in Osaka, March 23, 2022

14 / 31

Topics aboutcomplex and real
toric spaces

Introduction
Real toric spaces over simple polytopes

Zhang, SongAn equivalent way of construction
Introduction
Moment-angle
complexes andpolyhedral products

Each k-subspace H of (Z/2)m which acts freely on RZP uniquely determines a non-degenerate colouring
λ : (Z/2)m → (Z/2)m−k which is an epimorphism, and vice versa. Furthermore, under this situation,

Complex and real
toric spaces over
simple polytopes

Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Mλm(Z/2)m−k ∼=(Z/2) /H-equivariantly≈

RZP /H

Small covers: Introduced by Davis and Januszkiewicz in 1991, a special class of real toric spaces, for they
can be constructed by non-degenerate (Z/2)n -colourings, or equivalently, orbit spaces of RZP by free(Z/2)m−n -actions.

Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
The cubical decomposition of Pn

Zhang, Song

[Buchstaber and Panov, 2002 or Toric Topology]
Introduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P

Define cP : Pn → ∂Im = ∂[0, 1]m
by assigningthe barycentre of each proper face f = Fi1 ∩ · · · ∩ Fil 7→ (1 , . . . , m )where i = 0 if i ∈ {i1 , . . . , il }, otherwise i = 1,
the barycentre of Pn 7→ (1, . . . , 1)

A filtration of aquotient of
moment-angle
manifoldA cellular
decomposition of areal toric spaceand extending this assignment linearly on all the simplices of barycentric subdivision of Pn .

The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
The cubical decomposition of Pn

Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifold

NotationLet f = cP (f ) = {1} × · · · × [0, 1] × · · · × [0, 1] × · · · × {1},i1 -thil -thb f = {1} × · · · × [0, 1) × · · · × [0, 1) × · · · × {1}, and P = {1} × · · · × {1}. And topologically we consideri1 -thA cellular
decomposition of areal toric space

The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

il -ththe quotient of moment-angle manifold M(Pn , Λ) and the real toric space Mλ as (S1 )r × cP (Pn )/ ∼Λ and(Z/2)r × cP (Pn )/ ∼λ respectively, with the given equivalent relation(0y = y0 and yi1 = · · · = yil = 0, yi > 0 if i ∈/ {i1 , . . . , il }(ei2πt , y) ∼ (ei2πt , y0 ) ⇔0ei2πt · e−i2πt ∈ T (Λ(ei1 ),··· ,Λ(eil ))and(g, y) ∼ (g0 , y0 ) ⇔(y = y0 and yi1 = · · · = yil = 0, yi > 0 if i ∈/ {i1 , . . . , il }g0 − g ∈ Span{λ(ei1 ), . . . , λ(eil )}respectively.

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
A filtration of a quotient of moment-angle manifold M(Pn , Λ)Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifold

Given Λ : Zm → Zr , for each proper face f = Fi1 ∩ · · · ∩ Fil of Pn , choose column vectors αl+1 , . . . , αr ∈ Zr
such that Λf = λi1 , · · · , λil , αl+1 , · · · , αr has the determinant ±1. Let
Φf : (D2 )l × (S1 )r−l → π −1 (f ) = (S1 )r × f / ∼Λ , by
 i2πΛ ·t
f , 1, · · · , y , · · · , y , · · · , 1 .
yi1 ei2πt1 , · · · , yil ei2πtl , ei2πtl+1 , · · · , ei2πtr →
7
e
i1
il

A cellular
decomposition of areal toric space

It is easy to verify that

The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Φf is surjective, besides Φf(IntD2 )l ×(S1 )r−lb f ) = (S1 )r × 
b f / ∼Λ ,: (IntD2 )l × (S1 )r−l ≈ π −1 (for any face g of f such that cP (ĝ) = (1, · · · , 0 , · · · , 1 , · · · , 0 , · · · , 1),

Zhang, Song (Fudan University)

i1 -thil -th0

il -thΦf (D2 )l0 −1 × S1 × (D2 )l−l0 × (S1 )r−l = Φg (D2 )l−1 × (S1 )r−l+1 .

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
A filtration of a quotient of moment-angle manifold M(Pn , Λ)Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifold

Given Λ : Zm → Zr , for each proper face f = Fi1 ∩ · · · ∩ Fil of Pn , choose column vectors αl+1 , . . . , αr ∈ Zr
such that Λf = λi1 , · · · , λil , αl+1 , · · · , αr has the determinant ±1. Let
Φf : (D2 )l × (S1 )r−l → π −1 (f ) = (S1 )r × f / ∼Λ , by
 i2πΛ ·t
f , 1, · · · , y , · · · , y , · · · , 1 .
yi1 ei2πt1 , · · · , yil ei2πtl , ei2πtl+1 , · · · , ei2πtr →
7
e
i1
il

A cellular
decomposition of areal toric space

It is easy to verify that

The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Φf is surjective, besides Φf(IntD2 )l ×(S1 )r−lb f ) = (S1 )r × 
b f / ∼Λ ,: (IntD2 )l × (S1 )r−l ≈ π −1 (for any face g of f such that cP (ĝ) = (1, · · · , 0 , · · · , 1 , · · · , 0 , · · · , 1),

Zhang, Song (Fudan University)

i1 -thil -th0

il -thΦf (D2 )l0 −1 × S1 × (D2 )l−l0 × (S1 )r−l = Φg (D2 )l−1 × (S1 )r−l+1 .

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
A filtration of a quotient of moment-angle manifold M(Pn , Λ)Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifoldA cellular
decomposition of areal toric space

The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Thus we are able to endow a filtration to the manifold M(Pn , λ):where, for 0 ≤ l ≤ n, X r+l =X 0 ,→ · · · ,→ X r ,→ · · · ,→ X r+n = M(Pn , λ)SSΦf (D2 )l × (S1 )r−l =π −1 (f ); for 0 ≤ j ≤ r, X j is the j-th
dim f =n−l

dim f =n−lcellular skeleton of X r ≈ (S1 )r , which is decomposed canonically, i.e., take one 1-cell and one 0-cell {1} forS1 . Notice that formally, X 0 = {1} × · · · × {1} × (1, · · · , 1)/ ∼Λ and X r = (S1 )r × (1, · · · , 1)/ ∼Λ .

Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
A cellular decomposition of a real toric space MλZhang, SongDefinition: the f -vector of Pn [An Introduction to Convex Polytopes, Bronsted]
Introduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifoldf (P) = (f0 , · · · , fn−1 )is the number of i-faces of Pn . That is, fwhere fi
we let fn = 1.

n0 is the number of vertices of P , and fn−1 = m. In addition

Then given λ : (Z/2)m → (Z/2)r , we have

A cellular
decomposition of areal toric space

The cellular structure of MλThe cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

The n-dimensional CW-complex Mλ has l-cells with the number of fn−l · 2r−l , for l = 0, . . . , n.
(g)

Let Θ : (Z/2)r × Pn → Mλ , Gf =Span{λ(ei1 ), . . ., λ(eil )}, and Cf

(g)

Cf

=

S

h∈g+Gf Θ(h × f ). Then

≈ [−1, 1]n−dim f , for example:

Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
A cellular decomposition of a real toric space MλZhang, SongDefinition: the f -vector of Pn [An Introduction to Convex Polytopes, Bronsted]
Introduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifoldf (P) = (f0 , · · · , fn−1 )is the number of i-faces of Pn . That is, fwhere fi
we let fn = 1.

n0 is the number of vertices of P , and fn−1 = m. In addition

Then given λ : (Z/2)m → (Z/2)r , we have

A cellular
decomposition of areal toric space

The cellular structure of MλThe cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

The n-dimensional CW-complex Mλ has l-cells with the number of fn−l · 2r−l , for l = 0, . . . , n.
(g)

Let Θ : (Z/2)r × Pn → Mλ , Gf =Span{λ(ei1 ), . . ., λ(eil )}, and Cf

(g)

Cf

=

S

h∈g+Gf Θ(h × f ). Then

≈ [−1, 1]n−dim f , for example:

Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Filtrations of complex and real toric spaces
A cellular decomposition of a real toric space MλZhang, SongDefinition: the f -vector of Pn [An Introduction to Convex Polytopes, Bronsted]
Introduction
Filtrations ofcomplex and real
toric spaces
The cubicalndecomposition of P
A filtration of aquotient of
moment-angle
manifoldf (P) = (f0 , · · · , fn−1 )is the number of i-faces of Pn . That is, fwhere fi
we let fn = 1.

n0 is the number of vertices of P , and fn−1 = m. In addition

Then given λ : (Z/2)m → (Z/2)r , we have

A cellular
decomposition of areal toric space

The cellular structure of MλThe cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

The n-dimensional CW-complex Mλ has l-cells with the number of fn−l · 2r−l , for l = 0, . . . , n.
(g)

Let Θ : (Z/2)r × Pn → Mλ , Gf =Span{λ(ei1 ), . . ., λ(eil )}, and Cf

(g)

Cf

=

S

h∈g+Gf Θ(h × f ). Then

≈ [−1, 1]n−dim f , for example:

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

19 / 31

Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons
The induced spectral sequence by the filtration

Zhang, Song

Review

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Choose αl+1 , . . . , αr ∈ Zr such that |Λf = λi1 , · · · , λil , αl+1 , · · · , αr | = 1, andΦf : (D2 )l × (S1 )r−l → π −1 (f ) = (S1 )r × f / ∼ΛandX 0 ,→ · · · ,→ X r ,→ · · · ,→ X r+n = M(Pn , λ).

The induced
spectral sequence
The cohomology ofmoment-angle
manifolds overpolygons

The (co)homology of the pair (X r+l , X r+l−1 )There is an isomorphism(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

M
dim f =n−lFundamentalgroups of real
toric spaces over
simple polytopes

Φf ∗ :M ∼
=Hp (D2 )l × (S1 )r−l , ∂(D2 )l × (S1 )r−l → Hp X r+l , X r+l−1 ,∀p ≥ 0dim f =n−lThus we have the description
Hj (X r+l , X r+l−1 ) ∼= H j (X r+l , X r+l−1 ; Z) ∼
=







L



when 2l ≤ j ≤ r + l

dim f =n−l0,
Zhang, Song (Fudan University)r−l

Z j−2l ,when j < 2l or j > r + l

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Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons
The induced spectral sequence by the filtration

Zhang, Song

Review

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Choose αl+1 , . . . , αr ∈ Zr such that |Λf = λi1 , · · · , λil , αl+1 , · · · , αr | = 1, andΦf : (D2 )l × (S1 )r−l → π −1 (f ) = (S1 )r × f / ∼ΛandX 0 ,→ · · · ,→ X r ,→ · · · ,→ X r+n = M(Pn , λ).

The induced
spectral sequence
The cohomology ofmoment-angle
manifolds overpolygons

The (co)homology of the pair (X r+l , X r+l−1 )There is an isomorphism(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

M
dim f =n−lFundamentalgroups of real
toric spaces over
simple polytopes

Φf ∗ :M ∼
=Hp (D2 )l × (S1 )r−l , ∂(D2 )l × (S1 )r−l → Hp X r+l , X r+l−1 ,∀p ≥ 0dim f =n−lThus we have the description
Hj (X r+l , X r+l−1 ) ∼= H j (X r+l , X r+l−1 ; Z) ∼
=







L



when 2l ≤ j ≤ r + l

dim f =n−l0,
Zhang, Song (Fudan University)r−l

Z j−2l ,when j < 2l or j > r + l

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Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons
Betti numbers of ZP2Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
The induced
spectral sequence
The cohomology ofmoment-angle
manifolds overpolygons

Consider the special case of the quotient: ZP2 = M(P2 , id) itself, i.e., Λ : Zm → Zm is trivial. Set(
ΛF1 = (e1 , · · · , em ), . . . , ΛFm−1 = (em−1 , e1 , · · · , em−2 , em ), ΛFm = (em , e1 , · · · , em−1 );ΛF1 ∩F2 = (e1 , e2 , e3 , · · · , em ), . . . , ΛFm−1 ∩Fm = (em−1 , em , e1 , · · · , em−2 ), ΛF1 ∩Fm = (e1 , em , e2 , · · · , em−1 ).
Then the filtration X 0 ,→ · · · ,→ X m ,→ X m+1 ,→ X m+2 = ZP2 induce a spectral sequence which has theE1 -page:
δ

δ

1
1
1
0←Em+2,m+2←Em+1,m+1←Em,m0(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

001← · · ·←E4,4

0

1
←E3,3

0

1
←E2,2

0

0

1
1
←E1,1
←E0,0
←0

δ

1
1
←Em+1,m+2←Em,m+1
←0

..

.
δ

1
1
0 ←E4,m+2←E3,m+1←0

Fundamentalgroups of real
toric spaces over
simple polytopes

0

←0

1
←E2,m+1←01where, E∗,m+l= H ∗ X m+l , X m+l−1 ; Z .

Zhang, Song (Fudan University)

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Toric Topology 2022 in Osaka, March 23, 2022

21 / 31

Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons
Betti numbers of ZP2Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
The induced
spectral sequence
The cohomology ofmoment-angle
manifolds overpolygons

Consider the special case of the quotient: ZP2 = M(P2 , id) itself, i.e., Λ : Zm → Zm is trivial. Set(
ΛF1 = (e1 , · · · , em ), . . . , ΛFm−1 = (em−1 , e1 , · · · , em−2 , em ), ΛFm = (em , e1 , · · · , em−1 );ΛF1 ∩F2 = (e1 , e2 , e3 , · · · , em ), . . . , ΛFm−1 ∩Fm = (em−1 , em , e1 , · · · , em−2 ), ΛF1 ∩Fm = (e1 , em , e2 , · · · , em−1 ).
Then the filtration X 0 ,→ · · · ,→ X m ,→ X m+1 ,→ X m+2 = ZP2 induce a spectral sequence which has theE1 -page:
δ

δ

1
1
1
0←Em+2,m+2←Em+1,m+1←Em,m0(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

001← · · ·←E4,4

0

1
←E3,3

0

1
←E2,2

0

0

1
1
←E1,1
←E0,0
←0

δ

1
1
←Em+1,m+2←Em,m+1
←0

..

.
δ

1
1
0 ←E4,m+2←E3,m+1←0

Fundamentalgroups of real
toric spaces over
simple polytopes

0

←0

1
←E2,m+1←01where, E∗,m+l= H ∗ X m+l , X m+l−1 ; Z .

Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces
Zhang, SongThe cohomology of quotients of moment-angle manifolds over polygons
Betti numbers of ZP2

And the E2 - to Em -page:

Introduction
Filtrations ofcomplex and real
toric spaces

2
Em+2,m+2

The cohomology
of quotients ofmoment-angle
manifolds overpolygons

0

0

0

2
Em,m+1v δ

1Em−1,m−1
20..

The induced
spectral sequenceThe cohomology ofmoment-angle
manifolds overpolygons

Fundamentalgroups of real
toric spaces over
simple polytopes

1
E3,3

1
E2,2

1
E1,1

1
E0,0

δm−3
δm−2

.

0

(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

1
E4,4

···

δm−1



δm

2
E4,m+1002E3,m+100

1E2,m+1

0Thus we will finish an another proof of which was obtained by considering the E2 term of the Leray-Serre
spectral sequence for the fibre bundle T m−2 → ZP2 → M 4 :[Buchstaber and Panov, 1999]p

dim H (ZP2 ; Z) =Zhang, Song (Fudan University)((m − 2) m−2
−
p−2


m−2
− m−2
, if 3 ≤ p ≤ m − 1;p−1
p−30, if p = 1, 2, m, or m + 1;

Topics about complex and real toric spaces

1, if p = 0, or m + 2.
Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces
Zhang, SongThe cohomology of quotients of moment-angle manifolds over polygons
Betti numbers of ZP2

And the E2 - to Em -page:

Introduction
Filtrations ofcomplex and real
toric spaces

2
Em+2,m+2

The cohomology
of quotients ofmoment-angle
manifolds overpolygons

0

0

0

2
Em,m+1v δ

1Em−1,m−1
20..

The induced
spectral sequenceThe cohomology ofmoment-angle
manifolds overpolygons

Fundamentalgroups of real
toric spaces over
simple polytopes

1
E3,3

1
E2,2

1
E1,1

1
E0,0

δm−3
δm−2

.

0

(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

1
E4,4

···

δm−1



δm

2
E4,m+1002E3,m+100

1E2,m+1

0Thus we will finish an another proof of which was obtained by considering the E2 term of the Leray-Serre
spectral sequence for the fibre bundle T m−2 → ZP2 → M 4 :[Buchstaber and Panov, 1999]p

dim H (ZP2 ; Z) =Zhang, Song (Fudan University)((m − 2) m−2
−
p−2


m−2
− m−2
, if 3 ≤ p ≤ m − 1;p−1
p−30, if p = 1, 2, m, or m + 1;

Topics about complex and real toric spaces

1, if p = 0, or m + 2.
Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons
The cohomology ring of ZP2Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
The induced
spectral sequence

Also in [Buchstaber and Panov, 1999 and 2002 (Proposition 7.23)], they gave a description of thecohomology ring of ZP2 : the group H m+2 (ZP2 ) is one-dimensional with the fundamental class [v1 v2 u3 · · · um ]of the manifold as the generator, the product of two cohomology classes [vi1 uτ1 ] ∈ H 2+|τ1 | (ZP2 ) and[vi2 uτ2 ] ∈ H 2+|τ2 | (ZP2 ) equals [v1 v2 u3 · · · um ] (up to a sign) only when {{i1 }, {i2 }, τ1 , τ2 } is a partition of [m],and otherwise equals zero. Motivated by this, using a filtration-preserving approximation of the diagonal map,
An explicit presentation of H ∗ (ZP2 ; Z)

The cohomology ofmoment-angle
manifolds overpolygonsH ∗ (ZP2 ; Z) ∼=^[e, {eQ,la }Q,a ] R

where e is a generator of H m+2 (ZP2 ), and eQ,la is a generator of H 1+|Q| (ZP2 ). Let (*) denote thecomplementary condition of (Q, Q0 ), the ideal R is generated by the relations(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)0, if (Q, Q0 ) does not satisfy (*)

0, if (Q, Q0 ) satisfies (*) but a > a0 or a + 1 < a0eQ,la eQ0 ,l0 0 =a(−1)la −la−1 e, if (Q, Q0 ) satisfies (*) and a = a0


0
0

(−1)la+1 −la e, if (Q, Q0 ) satisfies (*) and a + 1 = a0

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Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons
The cohomology ring of ZP2Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
The induced
spectral sequence

Also in [Buchstaber and Panov, 1999 and 2002 (Proposition 7.23)], they gave a description of thecohomology ring of ZP2 : the group H m+2 (ZP2 ) is one-dimensional with the fundamental class [v1 v2 u3 · · · um ]of the manifold as the generator, the product of two cohomology classes [vi1 uτ1 ] ∈ H 2+|τ1 | (ZP2 ) and[vi2 uτ2 ] ∈ H 2+|τ2 | (ZP2 ) equals [v1 v2 u3 · · · um ] (up to a sign) only when {{i1 }, {i2 }, τ1 , τ2 } is a partition of [m],and otherwise equals zero. Motivated by this, using a filtration-preserving approximation of the diagonal map,
An explicit presentation of H ∗ (ZP2 ; Z)

The cohomology ofmoment-angle
manifolds overpolygonsH ∗ (ZP2 ; Z) ∼=^[e, {eQ,la }Q,a ] R

where e is a generator of H m+2 (ZP2 ), and eQ,la is a generator of H 1+|Q| (ZP2 ). Let (*) denote thecomplementary condition of (Q, Q0 ), the ideal R is generated by the relations(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)0, if (Q, Q0 ) does not satisfy (*)

0, if (Q, Q0 ) satisfies (*) but a > a0 or a + 1 < a0eQ,la eQ0 ,l0 0 =a(−1)la −la−1 e, if (Q, Q0 ) satisfies (*) and a = a0


0
0

(−1)la+1 −la e, if (Q, Q0 ) satisfies (*) and a + 1 = a0

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Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
The induced
spectral sequence
The cohomology ofmoment-angle
manifolds overpolygons(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

The cohomology of quotients of moment-angle manifolds over polygons(Co)homology groups of M(P2 , Λ) under a special class of colourings

What else is this filtration able to do? Given Λ : Zm → Zr such that all the 3 vectors{Λ(em ), Λ(e1 ), Λ(e2 )}, . . . , {Λ(em−1 ), Λ(em ), Λ(e1 )} from adjacent edges are dependent. SetΛF1 ∩F2 = (Λ1 , Λ2 , α1 , · · · , αr−2 ), · · · ,ΛFm−1 ∩Fm = (Λm−1 , Λm , α1 , · · · , αr−2 ), ΛF1 ∩Fm = (Λ1 , Λm , α1 , · · · , αr−2 );ΛF1 = ΛF1 ∩F2 , ΛF2 = ΛF2 ∩F3 , · · · , ΛFm = (Λm , Λ1 , α1 , · · · , αr−2 )where we choose α1 , · · · , αr−2 ∈ Zr such that | det ΛF1 ∩F2 | = · · · = | det ΛF1 ∩Fm | = 1.
Then the induced spectral sequence has the same form of E1 -page as before, and the E2 - to Er -page as:702
Er+2,r+2002
Er+1,r+2

2
Er,r+10

2Er,r+2d2

2d2 Er−1,r−1d302
Er−1,r+1

Zhang, Song (Fudan University)

H

2
E3,3

HI

2
E2,2

2
E1,1

I

K

2
E0,0

d3

0
..

Fundamentalgroups of real
toric spaces over
simple polytopes

2E4,4>· · ·

6 >

dr−2

dr−2
dr−1

.

dr−1
dr

2
E5,r+22
E4,r+1

0

0

2
E4,r+22
E3,r+10

0

2
E2,r+1

Topics about complex and real toric spaces

0

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Topics aboutcomplex and real
toric spaces
Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
The induced
spectral sequence
The cohomology ofmoment-angle
manifolds overpolygons(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons

The cohomology of quotients of moment-angle manifolds over polygons(Co)homology groups of M(P2 , Λ) under a special class of colourings

What else is this filtration able to do? Given Λ : Zm → Zr such that all the 3 vectors{Λ(em ), Λ(e1 ), Λ(e2 )}, . . . , {Λ(em−1 ), Λ(em ), Λ(e1 )} from adjacent edges are dependent. SetΛF1 ∩F2 = (Λ1 , Λ2 , α1 , · · · , αr−2 ), · · · ,ΛFm−1 ∩Fm = (Λm−1 , Λm , α1 , · · · , αr−2 ), ΛF1 ∩Fm = (Λ1 , Λm , α1 , · · · , αr−2 );ΛF1 = ΛF1 ∩F2 , ΛF2 = ΛF2 ∩F3 , · · · , ΛFm = (Λm , Λ1 , α1 , · · · , αr−2 )where we choose α1 , · · · , αr−2 ∈ Zr such that | det ΛF1 ∩F2 | = · · · = | det ΛF1 ∩Fm | = 1.
Then the induced spectral sequence has the same form of E1 -page as before, and the E2 - to Er -page as:702
Er+2,r+2002
Er+1,r+2

2
Er,r+10

2Er,r+2d2

2d2 Er−1,r−1d302
Er−1,r+1

Zhang, Song (Fudan University)

H

2
E3,3

HI

2
E2,2

2
E1,1

I

K

2
E0,0

d3

0
..

Fundamentalgroups of real
toric spaces over
simple polytopes

2E4,4>· · ·

6 >

dr−2

dr−2
dr−1

.

dr−1
dr

2
E5,r+22
E4,r+1

0

0

2
E4,r+22
E3,r+10

0

2
E2,r+1

Topics about complex and real toric spaces

0

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Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons(Co)homology groups of M(P2 , Λ) under a special class of colourings

Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Betti numbers of M(P2 , Λ)r−2

The induced
spectral sequenceaThe cohomology ofmoment-angle
manifolds overpolygonsb(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons





r−2

 

+(m−2)· p−2 +Hp (M(P2 , λ)) ∼= Z p−4
note that we define the binomial ab has the property that



r−2
p

, for 0 ≤ p ≤ r + 2= 0, if a < b or b < 0.

RemarkFundamentalgroups of real
toric spaces over
simple polytopes

Apparently it would be interesting if the Betti numbers of all the quotients of moment-angle manifolds overpolygons were given, however for general situations the differentials d would be too complicated to deal with,
it would be hard to represent their kernels and images, nevertheless there would be some hope to achive
this goal with this spectral sequence if we could find some way to simplify the calculation.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

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25 / 31

Topics aboutcomplex and real
toric spaces

The cohomology of quotients of moment-angle manifolds over polygons(Co)homology groups of M(P2 , Λ) under a special class of colourings

Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

Betti numbers of M(P2 , Λ)r−2

The induced
spectral sequenceaThe cohomology ofmoment-angle
manifolds overpolygonsb(Co)homology
groups of thequotients ofmoment-angle
manifold under
special colourings
over polygons





r−2

 

+(m−2)· p−2 +Hp (M(P2 , λ)) ∼= Z p−4
note that we define the binomial ab has the property that



r−2
p

, for 0 ≤ p ≤ r + 2= 0, if a < b or b < 0.

RemarkFundamentalgroups of real
toric spaces over
simple polytopes

Apparently it would be interesting if the Betti numbers of all the quotients of moment-angle manifolds overpolygons were given, however for general situations the differentials d would be too complicated to deal with,
it would be hard to represent their kernels and images, nevertheless there would be some hope to achive
this goal with this spectral sequence if we could find some way to simplify the calculation.

Zhang, Song (Fudan University)

Topics about complex and real toric spaces

Toric Topology 2022 in Osaka, March 23, 2022

25 / 31

Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes
Let F (P) be the set of facets,

Zhang, SongDefinition: the right-angled Coxeter group [Combinatorics of Coxeter Groups, Bjorner and Brenti]

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

WP =< sF : F ∈ F (P)|s2F = 1; (sF sF0 )2 = 1 if F ∩ F 0 6= ∅ >,note that sF sF0 has infinite order in WP if F ∩ F 0 = ∅.
[Davis and Januszkiewicz, 1991]

The Borel construction of the small cover Mλ , has the fundamental group
π1 (E(Z/2)n ×(Z/2)n Mλ ) ∼= WP .
Besides, the Borel construction gives the fibration Mλ → E(Z/2)n ×(Z/2)n Mλ → B(Z/2)n , which induces theshort exact sequenceφ

1 → π1 (Mλ ) → WP → (Z/2)n → 1
where φ(sF ) = λ(F).
Indeed, this is also the case for real toric spaces Mλ over simple polytopes, the fibration given by the Borel
construction also induces the exact sequenceφ

1 → π1 (Mλ ) → WP → (Z/2)r → 1
Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes
Let F (P) be the set of facets,

Zhang, SongDefinition: the right-angled Coxeter group [Combinatorics of Coxeter Groups, Bjorner and Brenti]

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

WP =< sF : F ∈ F (P)|s2F = 1; (sF sF0 )2 = 1 if F ∩ F 0 6= ∅ >,note that sF sF0 has infinite order in WP if F ∩ F 0 = ∅.
[Davis and Januszkiewicz, 1991]

The Borel construction of the small cover Mλ , has the fundamental group
π1 (E(Z/2)n ×(Z/2)n Mλ ) ∼= WP .
Besides, the Borel construction gives the fibration Mλ → E(Z/2)n ×(Z/2)n Mλ → B(Z/2)n , which induces theshort exact sequenceφ

1 → π1 (Mλ ) → WP → (Z/2)n → 1
where φ(sF ) = λ(F).
Indeed, this is also the case for real toric spaces Mλ over simple polytopes, the fibration given by the Borel
construction also induces the exact sequenceφ

1 → π1 (Mλ ) → WP → (Z/2)r → 1
Zhang, Song (Fudan University)

Topics about complex and real toric spaces

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes
However we still would like to study the fundamental group of the real toric space with some other methods,
which will lead to an explicit presentation of π1 (Mλ ):Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

When Mλ is a small cover [Wu and Yu, 2021](g)(g+λ(Fi ))(βFi )−1 = βFi
π1 (Mλ ) ∼
=

D

(g)
(g)
βF1 = · · · = βFn = 1,(g)βFi : g ∈ (Z/2)n , i = 1, . . . , m(g)(g+λ(Fi ))βFi · βFj

Fundamentalgroups of real
toric spaces over
simple polytopes(g+λ(Fj ))(g)= βFj · βFi;∀g ∈ (Z/2)n ;E, whenever Fi ∩ Fj 6= ∅Rewiew: cellular decomposition of Mλ(g)Let Θ : (Z/2)r × Pn → Mλ , Gf =Span{λ(ei1 ), . . ., λ(eil )}, and Cf=S

h∈g+Gf Θ(h × f ). Then(g)
Cf ≈ [−1, 1]n−dim f , which is the form of an l-cell. Note that(g)Cf(g+1 λ(Fi1 )+···+l λ(Fil ))= · · · = Cf(g+λ(Fi1 )+···+λ(Fil ))= · · · = Cf

, ∀1 , . . . , l ∈ {0, 1}(0,··· ,0)(1,··· ,1)(g), . . . , CP}, m · 2r−1 1-cells {CFi : i = 1, . . . , m, g ∈ (Z/2)r },(g)
fn−2 · 2r−2 2-cells {CF∩F0 : F ∩ F 0 6= ∅, g ∈ (Z/2)r }, and f0 · 2r−n n-cells
(g)
(g)
(h)
{Cv : v is a vertex of Pn , g ∈ (Z/2)r }, note that if g − h ∈/ Gv , Cv ∩ Cv = ∅.

Specially, there are 2r 0-cells {CPZhang, Song (Fudan University)

Topics about complex and real toric spaces

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes
However we still would like to study the fundamental group of the real toric space with some other methods,
which will lead to an explicit presentation of π1 (Mλ ):Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

When Mλ is a small cover [Wu and Yu, 2021](g)(g+λ(Fi ))(βFi )−1 = βFi
π1 (Mλ ) ∼
=

D

(g)
(g)
βF1 = · · · = βFn = 1,(g)βFi : g ∈ (Z/2)n , i = 1, . . . , m(g)(g+λ(Fi ))βFi · βFj

Fundamentalgroups of real
toric spaces over
simple polytopes(g+λ(Fj ))(g)= βFj · βFi;∀g ∈ (Z/2)n ;E, whenever Fi ∩ Fj 6= ∅Rewiew: cellular decomposition of Mλ(g)Let Θ : (Z/2)r × Pn → Mλ , Gf =Span{λ(ei1 ), . . ., λ(eil )}, and Cf=S

h∈g+Gf Θ(h × f ). Then(g)
Cf ≈ [−1, 1]n−dim f , which is the form of an l-cell. Note that(g)Cf(g+1 λ(Fi1 )+···+l λ(Fil ))= · · · = Cf(g+λ(Fi1 )+···+λ(Fil ))= · · · = Cf

, ∀1 , . . . , l ∈ {0, 1}(0,··· ,0)(1,··· ,1)(g), . . . , CP}, m · 2r−1 1-cells {CFi : i = 1, . . . , m, g ∈ (Z/2)r },(g)
fn−2 · 2r−2 2-cells {CF∩F0 : F ∩ F 0 6= ∅, g ∈ (Z/2)r }, and f0 · 2r−n n-cells
(g)
(g)
(h)
{Cv : v is a vertex of Pn , g ∈ (Z/2)r }, note that if g − h ∈/ Gv , Cv ∩ Cv = ∅.

Specially, there are 2r 0-cells {CPZhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes
As there exists 2r 0-cells in the CW-complex Mλ , it is hard to calculate the fundamental group directly. Thus
we need to do some deformation:

Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons

The first time of retract

There exists a CW-complex
M 0 , which is homotopically equivalent to the original manifold Mλ , has 2r−n0-cells, and (fn−l − nl ) · 2r−l l-cells for l = 1, . . . , n.

Fundamentalgroups of real
toric spaces over
simple polytopes

Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces
Zhang, Song

Fundamental groups of real toric spaces over simple polytopes
The second time of retract

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

There exists a CW-complex M̃, which is homotopically equivalent to the original manifold Mλ , has exactly
only one 0-cell, (m − n) · 2r−1 − 2r−n + 1 1-cells, and (fn−l − nl ) · 2r−l l-cells for l = 2, . . . , n.
We consider the 1-skeleton of M 0 as a graph, the vertices and edges of which are the 0-cells and 1-cells of
M 0 respectively. We obtain the needed CW-complex M̃ by extracting a maximal tree out of this graph.
RemarkIn the special case when r = n, i.e. Mλ is a small cover, M 0 has only one 0-cell, thus the 1-skeleton is just a
finite wedge of circles.
The fundamental group of the 1-skeleton M̃ 1 can be written as
π1 (M̃ 1 , Dv ) ∼=Z· · ∗ Z}| ∗ ·{z(m−n)·2r−1 −2r−n +1(g)(g+λ(F))(βF )−1 = βF
D

(g)
(g)
βF1 = · · · = βFn = 1,(g)∼
= βF : g ∈ (Z/2)r , F ∈ F (P)
(0)

(0)

Topics about complex and real toric spaces∀g ∈ (Z/2)r ;(λ(Fn+1 )+···+λ(Fr−1 ))βFn+1 = · · · = βFr = · · · = βFr
Zhang, Song (Fudan University);
E=1

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Topics aboutcomplex and real
toric spaces
Zhang, Song

Fundamental groups of real toric spaces over simple polytopes
The second time of retract

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

There exists a CW-complex M̃, which is homotopically equivalent to the original manifold Mλ , has exactly
only one 0-cell, (m − n) · 2r−1 − 2r−n + 1 1-cells, and (fn−l − nl ) · 2r−l l-cells for l = 2, . . . , n.
We consider the 1-skeleton of M 0 as a graph, the vertices and edges of which are the 0-cells and 1-cells of
M 0 respectively. We obtain the needed CW-complex M̃ by extracting a maximal tree out of this graph.
RemarkIn the special case when r = n, i.e. Mλ is a small cover, M 0 has only one 0-cell, thus the 1-skeleton is just a
finite wedge of circles.
The fundamental group of the 1-skeleton M̃ 1 can be written as
π1 (M̃ 1 , Dv ) ∼=Z· · ∗ Z}| ∗ ·{z(m−n)·2r−1 −2r−n +1(g)(g+λ(F))(βF )−1 = βF
D

(g)
(g)
βF1 = · · · = βFn = 1,(g)∼
= βF : g ∈ (Z/2)r , F ∈ F (P)
(0)

(0)

Topics about complex and real toric spaces∀g ∈ (Z/2)r ;(λ(Fn+1 )+···+λ(Fr−1 ))βFn+1 = · · · = βFr = · · · = βFr
Zhang, Song (Fudan University);
E=1

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Topics aboutcomplex and real
toric spaces
Zhang, Song

Fundamental groups of real toric spaces over simple polytopes
The second time of retract

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

There exists a CW-complex M̃, which is homotopically equivalent to the original manifold Mλ , has exactly
only one 0-cell, (m − n) · 2r−1 − 2r−n + 1 1-cells, and (fn−l − nl ) · 2r−l l-cells for l = 2, . . . , n.
We consider the 1-skeleton of M 0 as a graph, the vertices and edges of which are the 0-cells and 1-cells of
M 0 respectively. We obtain the needed CW-complex M̃ by extracting a maximal tree out of this graph.
RemarkIn the special case when r = n, i.e. Mλ is a small cover, M 0 has only one 0-cell, thus the 1-skeleton is just a
finite wedge of circles.
The fundamental group of the 1-skeleton M̃ 1 can be written as
π1 (M̃ 1 , Dv ) ∼=Z· · ∗ Z}| ∗ ·{z(m−n)·2r−1 −2r−n +1(g)(g+λ(F))(βF )−1 = βF
D

(g)
(g)
βF1 = · · · = βFn = 1,(g)∼
= βF : g ∈ (Z/2)r , F ∈ F (P)
(0)

(0)

Topics about complex and real toric spaces∀g ∈ (Z/2)r ;(λ(Fn+1 )+···+λ(Fr−1 ))βFn+1 = · · · = βFr = · · · = βFr
Zhang, Song (Fudan University);
E=1

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Topics aboutcomplex and real
toric spaces
Zhang, Song

Fundamental groups of real toric spaces over simple polytopes
The second time of retract

Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

There exists a CW-complex M̃, which is homotopically equivalent to the original manifold Mλ , has exactly
only one 0-cell, (m − n) · 2r−1 − 2r−n + 1 1-cells, and (fn−l − nl ) · 2r−l l-cells for l = 2, . . . , n.
We consider the 1-skeleton of M 0 as a graph, the vertices and edges of which are the 0-cells and 1-cells of
M 0 respectively. We obtain the needed CW-complex M̃ by extracting a maximal tree out of this graph.
RemarkIn the special case when r = n, i.e. Mλ is a small cover, M 0 has only one 0-cell, thus the 1-skeleton is just a
finite wedge of circles.
The fundamental group of the 1-skeleton M̃ 1 can be written as
π1 (M̃ 1 , Dv ) ∼=Z· · ∗ Z}| ∗ ·{z(m−n)·2r−1 −2r−n +1(g)(g+λ(F))(βF )−1 = βF
D

(g)
(g)
βF1 = · · · = βFn = 1,(g)∼
= βF : g ∈ (Z/2)r , F ∈ F (P)
(0)

(0)

Topics about complex and real toric spaces∀g ∈ (Z/2)r ;(λ(Fn+1 )+···+λ(Fr−1 ))βFn+1 = · · · = βFr = · · · = βFr
Zhang, Song (Fudan University);
E=1

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes

Zhang, Song

Whenever the given facets F ∩ F 0 6= ∅, they will determine a 2-cell of Mλ drawn as:
Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Thus, we obtain the relation set(g)(g+λ(F))R = {βF · βF0

Zhang, Song (Fudan University)(g+λ(F)+λ(F 0 ))· βF

(g+λ(F 0 ))· βF 0Topics about complex and real toric spaces: F ∩ F 0 6= ∅} ⊂ π1 (M̃ 1 , Dv )Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes

Zhang, Song

Whenever the given facets F ∩ F 0 6= ∅, they will determine a 2-cell of Mλ drawn as:
Introduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

Thus, we obtain the relation set(g)(g+λ(F))R = {βF · βF0

Zhang, Song (Fudan University)(g+λ(F)+λ(F 0 ))· βF

(g+λ(F 0 ))· βF 0Topics about complex and real toric spaces: F ∩ F 0 6= ∅} ⊂ π1 (M̃ 1 , Dv )Toric Topology 2022 in Osaka, March 23, 2022

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Topics aboutcomplex and real
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Fundamental groups of real toric spaces over simple polytopes

Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

The fundamental group of Mλπ1 (Mλ ) ∼= π1 (M̃, Dv ) ∼= π1 (M̃ 1 , Dv )/(normalization of R)(g)(g+λ(Fi ))(βFi )−1 = βFiD(g)∼= βFi : g ∈ (Z/2)r , i = 1, . . . , m(g)(g)
βF1 = · · · = βFn = 1,;∀g ∈ (Z/2)r ;E(λ(F)+···+λ(Fr−1 ))(0)(0)
βFn+1 = · · · = βFr = · · · = βFr n+1
= 1;
(g)

(g+λ(Fi ))βFi · βFj(g)(g+λ(Fj ))= βFj · βFi, whenever Fi ∩ Fj 6= ∅Thank you for your attention!

Zhang, Song (Fudan University)

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Topics aboutcomplex and real
toric spaces

Fundamental groups of real toric spaces over simple polytopes

Zhang, SongIntroduction
Filtrations ofcomplex and real
toric spaces
The cohomology
of quotients ofmoment-angle
manifolds overpolygons
Fundamentalgroups of real
toric spaces over
simple polytopes

The fundamental group of Mλπ1 (Mλ ) ∼= π1 (M̃, Dv ) ∼= π1 (M̃ 1 , Dv )/(normalization of R)(g)(g+λ(Fi ))(βFi )−1 = βFiD(g)∼= βFi : g ∈ (Z/2)r , i = 1, . . . , m(g)(g)
βF1 = · · · = βFn = 1,;∀g ∈ (Z/2)r ;E(λ(F)+···+λ(Fr−1 ))(0)(0)
βFn+1 = · · · = βFr = · · · = βFr n+1
= 1;
(g)

(g+λ(Fi ))βFi · βFj(g)(g+λ(Fj ))= βFj · βFi, whenever Fi ∩ Fj 6= ∅Thank you for your attention!

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