arXiv:1705.05531v1 [math.AT] 16 May 2017
A MOMENT-ANGLE MANIFOLD WHOSE COHOMOLOGY IS
NOT TORSION FREE
GEFEI WANG AND XIAOMENG LI
Abstract. In this paper we give a method to construct moment-angle manifolds whose cohomologies are not torsion free. We also give method to describe
the corresponding simplicial sphere by its non-faces.
1. Introduction
Corresponding to every abstract simplicial complex L on vertex set [m], there
are the real and complex moment-angle complexes RZL and ZL (cf. [3, 4]). They
are defined as
[Y
Y
1
RZL =
Di1 ×
Si0 ⊂ D11 × D21 × · · · × Dm
σ∈L i∈σ
ZL =
[Y
σ∈L i∈σ
i∈σ
/
Di2
×
Y
2
Si1 ⊂ D12 × D22 × · · · × Dm
.
i∈σ
/
The cohomology groups of RZL and ZL are given by Hochester theorem:
Theorem [1, 3, 4] Let L be a simpicial complex on vertex set [m], then
M
e ∗−1 (L|I )
H
H ∗ (RZL ) ∼
=
I⊂[m]
H (ZL ) ∼
=
∗
M
I⊂[m]
e ∗−|I|−1 (L|I )
H
where L|I is the full subcomplex of L on subset I and I rans over all the subsets
of [m].
From [6, 5, 7] we know that both RZL and ZL are topological manifolds if L
is a simplicial sphere, referred as moment-angle manifolds. Furthermore if L is
a polytopal sphere (the boundary complex of a simplicial polytope), then ZL is
a transverse intersection of real quadratic hypersurfaces (cf. [2]), while both ZL
and RZL are framed differentiable manifolds.
2010 Mathematics Subject Classification. 05A19, 05E40, 13F55, 52B05, 52B10.
Key words and phrases. moment-angle manifold, stellar subdivision, full subcomplex, missing
face and simplicial complement.
This project is supported by NSFC No.11471167.
1
2
G.WANG & X.LI
F.Bosio and L.Meersseman in [2] announced that the cohomology groups of
differentiable complex moment-angle manifolds may have any torsion Z/m. Furthermore if L is colourable, L.Cai, S.Choi and H. Park in [8, 9] proved that the
small cover under RZL may have any torsion Z/m.
From Hochester theorem, it is easy to construct a moment-angle complex whose
cohomology has torsion. But it is harder to construct such moment-angle manifolds (at least, the cohomology of all the moment-angle manifolds corresponding
to dimensional 1, 2 and 3 simplicial spheres are torsion free (cf. [2] Corollary
11.1)).
Based on Hochester theorem, our goal is to find a simplicial complex K whose
cohomology has torsion and K is embedded in a polytopal sphere L as a full
subcomplex. Then both the real and complex moment-angle complices corresponding to L are differentiable manifolds and the cohomology of RZL and ZL
e ∗ (K) as a summand.
have H
Theorem 3.2 (Construction) Let K be a subcomplex (not a full subcomplex)
of a simplicial sphere L0 in vertex set [m], M = {σ1 , σ2 , · · · , σs } be the set of
missing faces of K. As following, make stellar subdivisions at σ1 , σ2 , · · · , σs on
L0 one by one
L1 = ssσ1 L0 , L2 = ssσ2 L1 , · · · , Ls = ssσs Ls−1 .
Then K becomes a full subcomplex of Ls , K = Ls |[m] .
In fact every stellar subdivision on a polytopal sphere is still polytopal (see
[10]). If L0 is also a polytopal sphere, we thus obtain a polytopal sphere Ls
constructed by Theorem 3.2 such that K is a full subcomplex of Ls .
At last in section 4, we give a differentiable moment-angle manifold whose
cohomology has Z/3 as a summand. This is done as follows:
Triangulate the mod 3 Moore space K which has 8 vertices, 17 dimensional 2
facets and 26 missing faces (see figure 3). It could be embedded in a dimensional
4 polytopal sphere L0 . But here we only need to make 24 times of stellar subdivisions on it to make K becomes a full subcomplex of a polytopal sphere L24
because K and L0 has two common missing faces. Then L24 is a 4 dimensional
polytopal sphere with 32 vertices and H ∗ (ZL24 ) has H 2 (K) = Z/3 as a summand.
The Authors are grateful to professor Zhi. Lu for his helpful suggestion during
this study. This work was done under the supervise of professor Xiangjun. Wang.
2. Stellar Subdivision
An abstract simplicial complex K on vertex set [m] is a collection of simplices
that satisfies: for any simplex σ ∈ K all of its proper faces (proper subsets) are
simplices of K.
An abstract simplicial complex K could also be given by all of its non-faces
N = 2[m] − K
that satisfies: if σ ∈ N is NOT a simplex of K and σ ′ ⊇ σ then σ ′ ∈ N is not a
simplex of K.
TORSION
3
A simplex σ ∈ 2[m] is called a missing face (or minimal non-face) of K if it is
not a face of K, but all of its proper subsets are feces of K, i.e. ∂σ ⊂ K. An
abstract simplicial complex could also be given by its set of missing faces
M = {σ ∈ 2[m] |σ is a missing face of K}.
A subset σ ′ of [m] is NOT a simplex of K if and only if it contains a missing face
σ ∈ M as a subset.
It is easy to see that two simplicial complexes K and K ′ in vertex set [m] are
combinatorial equivalent if and only if their sets of missing faces M and M′ are
equivalent, i.e. there exists a one to one correspondence φ : [m] → [m] that gives
a one to one correspondence between M and M′ .
Definition 2.1 ([13] Definition 2.1) Let K be a simplicial complex in vertex
set [m] and M, N be the sets of missing faces and non-faces of K respectively.
We define a simplicial complement of K, denoted by
P = {σ1 , σ2 , · · · , σs },
to be a collection of non-faces that includes all the missing faces M i.e.
M ⊂ P ⊂ N.
Similar to the set of missing faces M, a subset σ of [m] is not a simplex of K
if and only if it contains a non-face σi in the simplicial complement P.
Definition 2.2 Let P, P′ be two simplicial complements, if they correspond
to the same simplicial complex, we say that P and P′ are equivalent denoted by
P ≃ P′ .
′
It is easy to see that two simplicial complements P and P are equivalent if
for every non-face σ ∈ P there exists a σ ′ ∈ P′ such that σ ′ ⊆ σ and for every
non-face σ ′ ∈ P′ there exists a σ ∈ P such that σ ⊆ σ ′ .
Let P = {σ1 , σ2 , · · · , σs } be a simplicial complement of K. For a non-face
σj ∈ P if there exists a σi ∈ P, i 6= j such that σi ⊆ σj , then we can remove σj
from P, the resulting simplicial complement
P′ = {σ1 , σ2 , · · · , σ
bj , · · · , σs }
is equivalent to P. In this case we call that P is reduced to P′ . Every simplicial
complement of K could be reduced to the set of missing faces by removing all
the larger non-faces.
Example 1 The simplicial complex K is composed by the maximal simplices
(1, 2),(2, 4), (3, 4) and their vertices in vertex set [4] (see Figure 1)
1
2
3
4
Figure 1
4
G.WANG & X.LI
The set of missing faces of K is M = {(1, 3), (1, 4), (2, 3)}. And
P = {σ1 = (1, 3), σ2 = (1, 4), σ3 = (2, 3), σ4 = (1, 3, 4), σ5 = (1, 4)}
is a simplicial complement of K where σ2 = (1, 4) = σ5 appeared twice. σ4 ⊃ σ1 ,
σ5 ⊇ σ2 so they could be removed from P to reduce to the set of missing faces M.
Remark: M = {(1), (2), · · · , (m)} is the set of missing faces of the empty
simplex {∅} (only the empty set is the simplex), which is different from the
empty complex ∅. M = {∅} is the set of missing faces of ∅, even the empty set ∅
is not a simplex.
Let K be a simplicial complex in vertex set [m] and σ be a simplex of K. The
link and star of σ are defined to be the simplicial complices
linkK σ ={τ ∈ K|σ ∪ τ ∈ K, σ ∩ τ = ∅},
starK σ ={τ ∈ K|σ ∪ τ ∈ K}.
The interior (open) star is defined to be
IntstarK σ = {τ ∈ K|σ ⊂ τ }
and the boundary of star is the simplicial complex
∂starK σ = starK σ − IntstarK σ = {τ ∈ K|σ ∪ τ ∈ K, σ 6⊂ τ }.
Lemma 2.3 Let P = {σ1 , σ2 , · · · , σs } be a simplicial complement of K. Then
P − σ = {σ1 − σ, σ2 − σ, · · · , σs − σ}
is a simplicial complement of linkK σ in vertex set [m] − σ. Furthermore if we
put P − σ in the vertex set [m], then it is a simplicial complement of starK σ.
Proof: From its definition, we know that a subset τ in vertex set [m] − σ is
NOT a simplex of linkK σ if and only if σ ∪ τ is not a simplex of K. In another
word, there exists a σi ∈ P such that σi ⊂ τ ∪ σ. This is equivalent to say that
σi − σ ⊂ τ ∪ σ − σ = τ .
Beside, if a simplex τ in vertex set [m] − σ contains a σi − σ, then σ ∪ τ ⊇ σi .
τ is not a simplex of linkK σ.
P − σ = {σ1 − σ, σ2 − σ, · · · , σs − σ}
is a simplicial complement of linkL σ in vertex set [m] − σ.
If we consider P − σ as a simplicial complement in vertex set [m], the corresponding simplicial complex is starK σ.
✷
Example 2 In Example 1, the link of the simplex (2) is two vertices linkK (2) =
{(1), (4)} and starK (2) is composed by two 1-simplices (1, 2), (2, 4).
M − (2) ={(1, 3) − (2), (1, 4) − (2), (2, 3) − (2)} = {(1, 3), (1, 4), (3)}
≃{(1, 4), (3)}
is a simplicial complement of linkK (2) in vertex set {1, 3, 4}. Put it in the vertex
set {1, 2, 3, 4}, it becomes the simplicial complement of starK (2).
Definition 2.4 Let P1 = {σ1 , σ2 , · · · , σs } and P2 = {τ1 , τ2 , · · · , τt } be the
simplicial complements of K1 and K2 in vertex set [m]. We define their join
P1 ∗ P2 to be
P1 ∗ P2 = {σi ∪ τj |σi ∈ P1 , τj ∈ P2 }.
TORSION
5
Lemma 2.5 Let K1 and K2 be two simplicial complices in vertex set [m],
P1 = {σ1 , σ2 , · · · , σs } and P2 = {τ1 , τ2 , · · · , τt } be the simplicial complements of
K1 and K2 respectively. Then
P1 ∗ P2 = {σi ∪ τj |σi ∈ P1 , τj ∈ P2 }
is a simplicial complement of K1 ∪ K2 in vertex set [m].
Proof: It is easy to see that a simplex σ in vertex set [m] is not a simplex
of K1 ∪ K2 if and only if it is not a simplex of neither K1 nor K2 . This implies
that there exists a σi ∈ P1 such that σi ⊆ σ and also exists a τj ∈ P2 such that
τj ⊆ σ. Thus σi ∪ τj ⊆ σ.
Besides, if a simplex σ contains a non-face σi ∪ τj ∈ P1 ∗ P2 , then σ is not a
simplex of neither K1 nor K2 . It is not a simplex of K1 ∪ K2 .
✷
Corollary 2.6 If the simplicial complement P is equivalent to P′ , then for any
simplex σ and simplicial complement P2
P − σ ≃P′ − σ,
P ∗ P2 ≃P′ ∗ P2 .
✷
Let σ be a simplex of a simplicial complex K. The stellar subdivision at σ on
K is defined to be the union of the simplicial complices K − IntstarK σ and the
cone cone∂starK σ along their boundary ∂starK σ, denoted by
ssσ K = (K − IntstarK σ) ∪ (cone∂starK σ)
After stellar subdivision, one more vertex is added which is the vertex of the
cone.
Theorem 2.7 Let P = {σ1 , σ2 , · · · , σs } be a simplicial complement of K. Then
{P, σ, (P − σ) ∗ (m + 1)} is a simplicial complement of ssσ K in vertex set [m + 1],
where
(P − σ) ∗ (m + 1) ={(σ1 − σ, m + 1), (σ2 − σ, m + 1), · · · , (σs − σ, m + 1)}.
Proof: From Lemma 2.3 we know that P − σ is a simplicial complement of
starK σ in vertex set [m]. A simplex τ in vertex set [m] is not in the boundary
∂starK σ = starK σ − IntstarK σ if and only if τ is not in the star or τ ∈ starK σ
but in the interior of star, i.e. there is a σi − σ such that τ ⊇ σi − σ or τ ⊇ σ.
Beside, if τ ⊇ σ, it might be not a simplex of K. If it is a simplex of K,
τ ∈ IntstarK σ. So
{σ, P − σ}
is a simplicial complement of ∂starK σ in vertex set [m].
Take the cone of ∂starK σ in vertex set [m + 1], a simplex (τ, m + 1) is not a
simplex of cone∂starK σ if and only if τ is not a simplex of ∂starK σ. So
{σ, P − σ}
is a simplicial complement of cone∂starK σ but in vertex set [m + 1].
A simplex τ in vertex set [m] is not a simplex of K − IntstarK σ if and only
if it is not a simplex of K or it is in IntstarK σ, i.e. there exists a σi ∈ P such
6
G.WANG & X.LI
that τ ⊇ σi or τ ⊇ σ. So {P, σ} is a simplicial complement of K − IntstarK σ in
vertex set [m].
Put the simplicial complex K − IntstarK σ in vertex set [m + 1], (m + 1) does
not appear in K. It is a ghost vertex and (m + 1) is a missing face. So
{P, σ, (m + 1)}
is a simplicial complement of K − IntstarK σ in vertex set [m + 1].
From Lemma 2.5, we know that {P, σ, (m + 1)} ∗ {σ, P − σ} is a simplicial
complement of ssσ K = (K − IntstarK σ) ∪ (cone∂starK σ), where


P ∗ {P − σ},
 P ∗ σ,

σ ∗ σ,
σ ∗ {P − σ},
{P, σ, (m + 1)} ∗ {σ, P − σ} =
.


(m + 1) ∗ σ, (m + 1) ∗ {P − σ}
Firstly
σ ∗ σ = σ ∈ {P, σ, (m + 1)} ∗ {σ, P − σ}.
Every subset σ ∪ τi ∈ P ∗ σ, (σ, m + 1) ∈ (m + 1) ∗ σ and σ ∪ (τi − σ) ∈ σ ∗ {P − σ}
contains σ. They could be removed from {P, σ, (m + 1)} ∗ {σ, P − σ}.
For any τi ∈ P, one has τi − σ ∈ P − σ. So
τi = τi ∪ (τi − σ) ∈ P ∗ {P − σ}.
Any other τi ∪ (τj − σ) ∈ P ∗ {P − σ} contains τi . They could be removed from
P ∗ {P − σ} and P ∗ {P − σ} is equivalent to P.
Thus {P, σ, (m + 1)} ∗ {σ, P − σ} could be reduced to
{P, σ, {P − σ} ∗ (m + 1)}
={P, σ, (σ1 − σ, m + 1), (σ2 − σ, m + 1), · · · , (σs − σ, m + 1)}
which is a simplicial complement of ssσ K.
✷
Remark: If σ is NOT a simplex of K, we still have {P, σ, {P − σ} ∗ (m + 1)}
as a simplicial complement of a simplicial complex. In that case, there exists a
σi ∈ P such that σi ⊆ σ. So σ could be removed from {P, σ, {P − σ} ∗ (m + 1)}
and σi − σ = ∅ ∈ P − σ. Thus (σi − σ, m + 1) = (m + 1) ∈ {P − σ} ∗ (m + 1)
and all the other (σj − σ, m + 1) could be removed from {P, σ, {P − σ} ∗ (m + 1)}.
That is to say that (m + 1) is a missing face and
{P, σ, {P − σ} ∗ (m + 1)} ≃ {P, (m + 1)}
is still a simplicial complement of K but a ghost vertex (m + 1) is added. We
still call it the stellar subdivision at σ on K.
Example In Example 1, we make stellar subdivision at σ = (2, 4) on K (see
Figure 2)
1
2
1
2
ss
σ
=⇒
3
4
5
3
Figure 2
4
TORSION
7
P = {(1, 3), (1, 4), (2, 3)} is a simplicial complement of K, σ = (2, 4)
{P − σ} ∗ (5) = {(1, 3), (1), (3)} ∗ (5) ={(1, 3, 5), (1, 5), (3, 5)}.
{P, σ, {P − σ} ∗ (5)} = {(1, 3), (1, 4), (2, 3), (2, 4) = σ, (1, 3, 5), (1, 5), (3, 5)}
is a simplicial complement of ssσ K.
3. Construction
After given the simplicial complement of stellar subdivision we construct our
moment-angle manifolds whose cohomology has torsion.
Lemma 3.1 Let K be a simplicial complex and P = {σ1 , σ2 , · · · , σs } be a
simplicial complement of it. Let I be a subset of the vertex set [m]. Then
P|I = {σi ∈ P|σi ⊂ I}
is a simplicial complement of the full subcomplex K|I in vertex set I.
Proof: From its definition, we know that the full subcomplex
K|I = {σ ∈ K|σ ⊂ I}
is a simplicial complex in vertex set I. A subset τ in vertex set I is not a simplex
of K|I if and only if τ is not a simplex of K. In another words, there exists a
non-face σi ∈ P such that τi ⊂ τ . Note that τ ⊂ I, τi ⊂ τ ⊂ I. The Lemma
follows.
✷
Theorem 3.2 (Construction) Let K be a subcomplex (not a full subcomplex)
of a simplicial sphere L0 in vertex set [m], M = {σ1 , σ2 , · · · , σs } be the set of
missing faces of K. As following, make stellar subdivisions at σ1 , σ2 , · · · , σs on
L0 one by one
L1 = ssσ1 L0 , L2 = ssσ2 L1 , · · · , Ls = ssσs Ls−1 .
Then K becomes a full subcomplex of Ls , K = Ls |[m] .
Proof: Let P0 = {τ1 , τ2 , · · · , τr } be a simplicial complement of L0 . From
Theorem 2.7 we know that
P1 = {P0 , σ1 , P′1 }
is a simplicial complement of L1 = ssσ1 L0 where P′1 = {P0 − σ1 } ∗ (m + 1).
′
P2 ={P1 , σ2 , P2 } = {P0 , σ1 , σ2 , P′1 , P′2 }
′
is a simplicial complement of L2 = ssσ2 L1 with P2 = (P1 − σ2 ) ∗ (m + 2).
By induction, we get a simplicial complement of Ls = ssσs Ls−1 as
Ps ={Ps−1 , σs , P′s }
={P0 , σ1 , σ2 , · · · , σs , P′1 , P′2 , · · · , P′s }
with P′i = {Pi−1 − σi } ∗ (m + i). Every non-face in P′i contains m + i as a vertex.
Thus
Ps |[m] = {P0 , σ1 , σ2 , · · · , σs }
8
G.WANG & X.LI
is a simplicial complement of the full subcomplex Ls |[m] . Note that K is a subcomplex of L0 , every non-face τi ∈ P0 is not a simplex of K. There exists a
σj ∈ M such that σj ⊆ τi and τi could be removed from {P0 , σ1 , σ2 , · · · , σs }.
Ps |[m] = {P0 , σ1 , σ2 , · · · , σs } ≃ {σ1 , σ2 , · · · , σs }
which is the set of missing faces of K.
✷
In fact, if L0 is also a polytopal sphere, the stellar subdivision of it is also
polytopal. It has been proved in a geometric sense by G. Ewald and G. C.
Shephard in [10].
e 0 be the simplicial polytope and its boundary ∂ L
e 0 = L0 be the polytopal
Let L
sphere. If σ is a simplex of L0 and σ is the intersection of the facets (maximal
simplices of L0 ) Fi1 , Fi2 , · · · , Fir , one can take any point p beyond the facets
Fi1 , Fi2 , · · · , Fir and beneath the other facets (See [11] p.78 for the definitions
e 0 is the boundary of the
of beyond and beneath). The stellar subdivision ssσ ∂ L
′
e
e
convex hall of L0 = conv(L0 ∪ p).
It could also be proved from the duality of polytopes.
e 0 be the simplicial polytope corresponding to L0 , and P be the dual
Let L
simple polytope, (the vertex of L0 corresponding to the facet while the facet of
L0 corresponding to the vertex of P ). Let σ = (i1 , i2 , · · · , ik ) be a simplex of
L0 , make a stellar subdivision at σ on L0 is equivalent, though the duality of
polytopes, to cutting off the face σ ∗ = Fi1 ∩ Fi2 ∩ · · · ∩ Fik in P by a generic
hyperplane. The cutting off operation on a simple polytope is still simple, so
e 0 is polytopal.
ssσ ∂ L
4. Example
Let K be the triangulated mod 3 Moore space (see Figure 3) which could be
embedded in dimensional 4 polytopal sphere
L0 = Σ2 (∂∆3 ) = ∂(1, 2, 3, 4) ∗ ∂(5, 7) ∗ ∂(6, 8).
The set of missing faces of L0 is
P0 = {(1, 2, 3, 4), (5, 7), (6, 8)}.
The set of missing faces of K is

(1, 2, 3), (1, 2, 6), (1, 2, 7), (1, 2, 8), (1, 3, 4), (1, 3, 7),




 (1, 4, 5), (1, 4, 6), (1, 4, 7), (1, 5, 6), (1, 7, 8),
(2, 3, 5), (2, 3, 8), (2, 4, 5), (2, 4, 6), (2, 4, 7), (2, 4, 8), (2, 6, 7),
M=


(3, 4, 6), (3, 4, 7), (3, 4, 8), (3, 5, 6), (3, 6, 7)


 (5, 7), (6, 8), (5, 8),











where (5, 7) and (6, 8) are also the missing faces of L0 . So we only need to make
24 times stellar subdivisions at the other missing faces of K on it. We thus
obtain a dimensional 4 polytopal sphere L24 with 32 vertices which has K as
a full subcomplex. The real moment-angle manifold corresponding to L24 is of
dimensional 4 while the the complex one is of dimensional 37 where H 3 (RZL24 )
and H 11 (ZL24 ) has H 2 (K) = Z/3 as a summand.
TORSION
9
1
3
2
8
2
4
7
3
5
6
1
3
1
2
Figure 3
Compute the missing faces after making stellar subdivision at (1, 2, 3) and
(1, 2, 6) on L0 in different sequence, one has
(1) We make stellar subdivision at (1, 2, 3) on L0 at first, then make stellar
subdivision at (1, 2, 6).
From Theorem 2.7 we know that,
o
n
P0 = (1, 2, 3, 4), (5, 7), (6, 8)
σ1 =(1, 2, 3)
n
o
(P0 − σ1 ) ∗ (9) = (4, 9), (5, 7, 9), (6, 8, 9)
is a simplicial complement of L1 = ss(1,2,3) L0 . After removing the larger
non-faces (1, 2, 3, 4), (5, 7, 9) and (6, 8, 9) we get the set of missing faces
of L1
M1 = {(1, 2, 3), (5, 7), (6, 8), (4, 9)} .
Then we make stellar subdivision at (1, 2, 6) on L1 and get the set of
missing faces of L2
M2 = {(1, 2, 3), (1, 2, 6), (5, 7), (6, 8), (4, 9), (3, 10), (8, 10)} .
(2) Similarly we make stellar subdivision at (1, 2, 6) on L0 at first, then make
stellar subdivision at (1, 2, 3), the resulting set of missing faces of L′2 is
M′2 = {(1, 2, 3), (1, 2, 6), (5, 7), (6, 8), (3, 4, 9), (8, 9), (4, 10), (6, 10)} .
Comparing with these two sequence, we can find that even the number of
missing faces of L2 and L′2 are different. L2 has 7 missing faces while L′2 has 8. It
means that L2 is NOT combinatorial isomorphic to L′2 and this difference might
keep all the way during the later construction!
Remark: Though K will be a full subcomplex of Ls in every sequence of
making stellar subdivisions at K’s missing faces, the combinatorial structure of
Ls may NOT be combinatorial isomorphic in different sequences.
10
G.WANG & X.LI
References
[1] I. V. Baskakov, Cohomology of K-powers of spaces and the combinatorics of simplicial
divisions, Uspekhi Mat. Nauk 57 (2002), no. 5, 147–148 (Russian), Russian Math. Surveys,
57(2002), no. 5, 898-990 (English translation).
[2] F. Bosio and L. Meersseman, Real quadrics in Cn , complex manifolds and convex polytopes,
Acta Math. 197 (2006), no. 1, 53–127.
[3] V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and
combinatorics, University Lecture Series, vol. 24, Amer. Math. Soc., Providence, RI, 2002.
[4] V. M. Buchstaber and T. E. Panov, Toric topology, Mathematical Surveys and Monographs,
vol. 204, Amer. Math. Soc., Providence, RI, 2015.
[5] V. M. Buchstaber, T. E. Panov and N. Ray, Spaces of polytopes and cobordism of quasitoric
manifolds, Mosc. Math. J. 7 (2007), no. 2, 219–242.
[6] V. M. Buchstaber and N. Ray, Tangential structures on toric manifolds, and connected
sums of polytopes. Internat. Math. Res. Notices (2001), no. 4, 193–219.
[7] L. Cai, Norm minima in certain Siegel leaves, Algebr. Geom. Topol. 15 (2015), no. 1,
445–466.
[8] L. Cai and S. Choi, On the topology of a small cover associated to a shellable complex,
preprint (2016) arXiv:1604.06988
[9] S. Choi and H. Park, On the cohomology and their torsion of real toric objects, preprint
(2013); arXiv:1311.7056
[10] G. Ewald and G. C. Shephard, Stellar Subdivisions of Boundary Complexes of Convex
Polytopes, Math. Ann. 210, 7–16(1974)
[11] B. Grünbaum, Convex Polytopes, Graduate Texts in Mathematics 221 Springer-Verlag
2003.
[12] J. R. Munkres, Elements of Algebraic Topology, Addision-Wesley Publishing Company 1984.
[13] X. Wang and Q. Zheng, The homology of simplicial complements and the cohomology of
polyhedral products, 27 (2015), no. 4, 2267-2299.
School of Mathematical Science, Nankai University, Tianjin 300071, P. R. China
E-mail address: [email protected]
E-mail address: [email protected]