A Golod complex with non-suspension moment-angle complex

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by  Kouyemon Iriye

A GOLOD COMPLEX WITH NON-SUSPENSION MOMENT-ANGLE
COMPLEX
KOUYEMON IRIYE AND TATSUYA YANO
6
1
0
2
Abstract. It could be expected that the moment-angle complex associated with a Golod
b
e simplicial complex is homotopy equivalent to a suspension space. In this paper, we provide a
F counter example to this expectation. We have discovered this complex through the studies of
8 the Golodpropertyofthe Alexanderdualofajoinofsimplicialcomplexes,andthatofa union
of simplicial complexes.

]
T
A
1. Introduction
.
h
t The Stanley-Reisner ring (or face ring) of a simplicial complex K over an index set [m] =
a
m
{1,··· ,m} is defined as the quotient graded algebra
[

k[K] = k[v ,··· ,v ]/I ,
3 1 m K
v
2 where k is a commutative ring with unit and I = (v ···v | {i ,··· ,i } 6∈ K) is the Stanley-
7 K i1 ik 1 k
Reisner ideal of K. K is called Golod over a field k if its Stanley-Reisner ring k[K] is Golod
4
3 over k. That is, the multiplication and all higher Massey products in Tork[v1,···,vm](k[K],k) =
0
1. H(Λ[u1,··· ,um]⊗k[K],d) are trivial, where the Koszul differential algebra (Λ[u1,··· ,um]⊗
0 k[K],d) is the bigraded differential algebra with degui = (1,2), degvi = (0,2), and dui = vi
6
for i = 1,··· ,m. Originally, the algebra k[K] or the ideal I was defined to be Golod if the
1 K
: following equation holds:
v
i
arX i≥X0; j≥0dimkTorjk,[2Ki](k,k)tjzi = 1−tPi≥0; j≥1dim(k1T+otrzkj,)[2vni1,···,vm](k[K],k)tjzi,
where Tork[K](k,k) and Tork[v1,···,vm](k[K],k) denote the homogeneous components of degree
j,2i j,2i
2i. Golod [8] proved the equivalence of the two conditions, and thereafter his name has been
used to refer a ring that satisfies the condition. The reader may also refer to Gulliksen and
Levin [10] or Avramov [1].
Baskakov, Buchstaber, and Panov [3] and Franz [7] independently demonstrated that the
torsion algebra Tork[v1,···,vm](k[K],k) is isomorphic to the cohomology ring of the moment-angle
complex Z associated with K.
K
2010 Mathematics Subject Classification. 55P15(13F55).
Key words and phrases. Stanley-Reisner ring, Golod, Massey product, Hopf map.
K.I. is supported by JSPS KAKENHI (No. 26400094).
1
2 KOUYEMONIRIYEANDTATSUYAYANO
Theorem 1.1([3, 7]). For a commutative ring k with unit, the following isomorphisms of
algebras hold:
H∗(ZK;k) ∼= Tork∗[v1,···,vm](k[K],k) ∼= M H˜∗(KI;k),
I⊂[m]
where H˜∗(K ;k) denotes the reduced cohomology of the full subcomplex K of K on I, and
I I
H˜∗(K ;k) = 0 for ∗ =6 −1 and = k for ∗ = −1. The last isomorphism is the sum of isomor-
∅
phisms given by
Hp(ZK;k) ∼= M H˜p−|I|−1(KI;k),
I⊂[m]
and the ring structure is given by the maps
H˜p−|I|−1(K ;k)⊗H˜q−|J|−1(K ;k) → H˜p+q−|I|−|J|−1(K ;k)
I J I∪J
that are induced by the canonical simplicial maps K → K ∗ K for I ∩ J = ∅ and zero
I∪J I J
otherwise, where K ∗K denotes the join of two simplicial complexes K and K .
I J I J
Here, we recall that if the moment-angle complex Z is (homotopy equivalent to) a sus-
K
pension, then the multiplication and all higher Massey products in H∗(Z ;k) are trivial. For
K
example, see Corollary 3.11 of [22]. That is, the following implication holds:
(1.1) Z is a suspension =⇒ K is Golod,
K
where K is Golod if K is Golod over any field k. This observation enables us to investigate the
Golod property through the study of moment-angle complexes. One of the first studies in this
direction was introduced by Grbi´c and Theriault [9]. They demonstrated that the moment-
angle complex associated with a shifted simplicial complex is homotopy equivalent to a wedge
of spheres. In [14], Kishimoto and the first author extended this result to dual sequentially
Cohen-Macaulay complexes, and provided some new Golod complexes. In these studies, the
following theorem concerning the decomposition of polyhedral products (see Definition 2.1), as
introduced by Bahri, Bendersky, Cohen, and Gitler [2], plays an essential role.
Theorem 1.2([2]). Let K be a simplicial complex on an index set [m] and let (CX,X) =
{(CX ,X )} , where each X is a based space and CX is the reduced cone of a based space
i i i∈[m] i
X. Then, the following homotopy equivalence holds:
ΣZK(CX,X) ≃ Σ _ Σ|KI|∧XI,
b
I⊂[m]
where XI = ∧ X and X∅ = ∗.
i∈I i
b b
We call this decomposition of polyhedral products the BBCG decomposition for K. If this
decomposition is desuspendable, i.e., if the homotopy equivalence
ZK(CX,X) ≃ _ Σ|KI|∧XI
b
I⊂[m]
A GOLOD COMPLEX WITH NON-SUSPENSION MOMENT-ANGLE COMPLEX 3
holds for any sequence of based CW-complexes X, then we say that the BBCG decomposition
is desuspendable for K.
In this paper, we study the Golod properties of the Alexander dual of K ∗ L and K ∪ L,
α
where α is a common face of K and L. The precise statements of the results are given in the
next section.
By Theorem 1.1, the multiplicative structure of H∗(Z ;k) is trivial if and only if the maps
K
K → K ∗ K for I ∩ J = ∅ induce the trivial maps on the reduced cohomology theory.
I∪J I J
By strengthening this condition, K is said to be (stably) homotopy Golod [14] if the maps
|K | → |K ∗K | for I ∩J = ∅ are (stably) null homotopic and H∗(Z ;k) has trivial higher
I∪J I J K
Massey products for any fields k. By definition, the following implication holds:
(stably) homotopy Golod =⇒ Golod.
The second purpose of this paper is to prove that this implication is strict.
Theorem 1.3. There is a Golod simplicial complex K such that K is not stably homotopy
Golod. Moreover, Z can be chosen to be torsion free.
K
Here, a space or a simplicial complex X is called torsion free if its integral homology groups
H (X;Z) are torsion free.
∗
It could be expected that the converse of the implication (1.1) is also true. Theorem 1.3
provides a counter example to this expectation. In fact, if Z is a suspension then the fat
K
wedge filtration of Z is trivial, by Theorem 1.3 of [14]. By Theorem 6.9 of the same paper,
K
we see that K is stably homotopy Golod, which contradicts our result. Thus, Z is not a
K
suspension.
In the next section, we state the main results of this paper. The subsequent sections are
devoted to their proofs.
We are grateful to Daisuke Kishimoto for providing useful discussions and pointing out the
paper [18], which was applied in the construction of K in Theorem 1.3. We also thank Lukas
Katth¨an for his comments on the first draft of this paper. He kindly pointed out the papers
[12] and [6], and provided us with ideas for simplifying the arguments in the subsection 5.3.
2. Results
In this section, we state our main results. We begin by setting notation regarding simplicial
complexes.
Let K be a simplicial complex on an index set V. In this paper, we only consider finite
simplicial complexes. The subset V(K) of V defined by V(K) = ∪ σ is called the vertex
σ∈K
set of K, and an element of V −V(K) is called a ghost vertex. For a finite set V, we denote
the full simplex on V by ∆V. Its boundary is denoted by ∂∆V. We also use the symbol ∆n
to denote an n-dimensional simplex. |K| denotes a geometric realization of K. The link and
4 KOUYEMONIRIYEANDTATSUYAYANO
star of a face σ of K are denoted by link (σ) and star (σ), respectively. For a subset I ⊂ V,
K K
K denotes the full subcomplex of K indexed by I; that is, K = {σ ⊂ I | σ ∈ K}. Then, K
I I I
is called the restriction of K to I. For a vertex v of K, the deletion of v from K is denoted
by K −v = K . For simplicial complexes K and L with disjoint index sets V and W, the
V−{v}
simplicialjoinK∗LontheindexsetV⊔W isdefinedbyK∗L = {σ⊔τ ⊂ V⊔W |σ ∈ K, τ ∈ L},
where ⊔ always denotes the disjoint union of sets. We write K ∗c to stand for K ∗∆{c}. The
(simplicial) Alexander dual K∨ is defined by K∨ = {σ ⊂ V | σc = V −σ 6∈ K}. If V = V(K)
V V
or V is clear from a context, we simply write K∨ for K∨. It is easy to see that for a subset I
V
of V, I is a facet of K if and only if Ic is a minimal-non face of K∨. The restriction of K∨ and
the link of K are related by the following formula: (K∨) = (link (Ic))∨.
I K
Nextwereviewpolyhedralproducts, whichareageneralizationofamoment-anglecomplexes.
Definition 2.1. Let K be a simplicial complex on [m], and (X,A) be a sequence of pairs of
based spaces {(X ,A )} . The polyhedral product Z (X,A) is defined by
i i i∈[m] K
ZK(X,A) = [(X,A)σ (⊂ X1 ×···×Xm),
σ∈K
where (X,A)σ = Y ×···×Y , with Y = X for i ∈ σ and A for i 6∈ σ. If (X ,A ) = (X,A)
1 m i i i i i
for all i ∈ [m], then we write Z (X,A) for Z (X,A).
K K
Example 2.2. In [21], Porter used a polyhedral product to define a higher order Whitehead
product, and proved the following homotopy equivalence:
m
Z∂∆[m](CX,X) = [CX1 ×···×CXi−1 ×Xi ×CXi+1 ×···×CXm
i=1
≃ X ∗···∗X
1 m
≃ Σm−1X ∧···∧X .
1 m
Thepolyhedral product Z (D2,S1) is themoment-angle complex of K, andiswritten simply
K
as Z . We refer the reader to [14, 5] for further examples of polyhedral products. In this paper,
K
we are interested in the homotopy types of the polyhedral products Z (CX,X).
K
Theorem 2.3. Let K and L be simplicial complexes on disjoint index sets V and W,
respectively. If K 6= ∆V and L 6= ∆W, then the BBCG decomposition is desuspendable for
(K ∗L)∨.
Corollary 2.4. Let K and L be simplicial complexes on disjoint index sets V and W,
respectively. If K 6= ∆V and L 6= ∆W, then (K ∗L)∨ is Golod.
Because the Stanley-Reisner ideal of (K ∗ L)∨ is the product of those of K∨ and L∨, this
corollary provides a topological proof of a classical result given in [12]. The reader may also
A GOLOD COMPLEX WITH NON-SUSPENSION MOMENT-ANGLE COMPLEX 5
refer to [6] and [16]. We also remark that (∆V ∗L)∨ is Golod if and only if L∨ is Golod, because
(∆V ∗L)∨ = ∆V ∗L∨ (see Lemma 3.1).
Theorem 2.5. Let K and L be simplicial complexes of non-negative dimension on index
sets V and W, respectively. Assume that α = V ∩W is a common face of K and L. If K 6= ∆V
or L 6= ∆W, and α is not a facet of K or L, then the BBCG decomposition is desuspendable
for (K ∪ L)∨.
α
Corollary 2.6. Let K and L be simplicial complexes that satisfy the same conditions as
stated in Theorem 2.5. If α is not a facet of K or L, then (K ∪ L)∨ is Golod.
α
It is natural to ask whether (K ∪ L)∨ is still Golod even when α is a facet of K or L in
α
Corollary 2.6. In general, the answer is that this does not hold. Whether it does depend on K,
L, and α. In fact, we can construct many simplicial complexes (∆V ∪ L)∨ that are not Golod,
α
such as in Example 5.3.4. Incidentally, (∆V ∪ ∆W)∨ is a non-Golod simplicial complex, where
α
(∆V ∪ ∆W)∨ = ∂∆V−α ∗∆α ∗∂∆W−α (see Lemma 4.1).
α
To construct a simplicial complex satisfying Theorem 1.3, we first need to fix some notation.
Let K be a simplicial complex on an index set V, with facets F ,··· ,F . We take new
1 k
vertices v ,··· ,v , and define a new simplicial complex F(K) on the index set V ⊔{v ,··· ,v }
1 k 1 k
with facets F +v ,··· ,F +v , where F +v = F ⊔{v } as usual. Then, K is a subcomplex
1 1 k k i i i i
of F(K) and |K| is a deformation retract of |F(K)|.
For two simplicial complexes K and L, we define a “product” K⊠L as follows. Define a liner
order ≤ on the vertex sets of K and L. The vertex set of K⊠L is V(K)×V(L). An n-simplex
is a set {(x ,y ),··· ,(x ,y )} such that x ≤ ··· ≤ x , y ≤ ··· ≤ y , {x ,··· ,x } is a simplex
0 0 n n 0 n 0 n 0 n
of K, and {y ,··· ,y } is a simplex of L. It is well-known that |K ⊠ L| is homeomorphic to
0 n
|K|×|L|. If v is a vertex of L, then the subcomplex K⊠∆{v} of K⊠L is abbreviated as K⊠v.
It follows from the simplicial approximation theorem that there is a simplicial map η : S3 →
k k
S2 whose geometrical realization |η | : |S3| → |S2| is homotopic to the Hopf map η : S3 → S2,
4 k k 4
where Sn denotes a triangulation of an n-sphere Sn with k-vertices. In fact, we can choose
k
k = 12 in this case, by [18].
We consider the simplicial set ∆1 as the full simplex on [2]. By S3⊠∆1∪ S2 we denote the
k ηk 4
simplicial complex obtained from the disjoint union of two simplicial complexes S3 ⊠∆1 ⊔S2,
k 4
which is defined by identifying (v,2) ∈ V(S3) × [2] with η (v) ∈ V(S2). We embed S3 into
k k 4 k
S3 ⊠∆1 ∪ S2 by applying the map v 7→ (v,1).
k ηk 4
We set α to be the vertex set of (S3 ⊠ ∆1 ∪ S2) ∪ F(S3). Finally, we take new vertices
k ηk 4 k
v ,w ,w and set
0 1 2
K = ∆α+v0 ∪((S3 ⊠∆1 ∪ S2)∪F(S3))∗∆{w1} ∪S3 ∗∆{w1,w2},
k ηk 4 k k
which is the union of ∆α+v0 and ∆α ∪((S3 ⊠∆1 ∪ S2)∪F(S3))∗∆{w1} ∪S3 ∗∆{w1,w2}.
k ηk 4 k k
6 KOUYEMONIRIYEANDTATSUYAYANO
Theorem 2.7. K∨ is a Golod simplicial complex that is not stably homotopy Golod. More-
over, Z is torsion free if K is constructed from the map η : S3 → S2 defined in [18].
K∨ 12 12 4
3. Proof of Theorem 2.3.
In this section, we prove Theorem 2.3. We begin by stating some elementary lemmas, for
which the proofs are omitted.
Lemma 3.1. Let K and L be simplicial complexes with disjoint index sets V and W,
respectively. Then, (K ∗L)∨ = K∨ ∗∆W ∪∆V ∗L∨ .
V⊔W V W
Lemma 3.2. Let K and L for i = 1,2 be simplicial complexes with disjoint index sets.
i i
Then, (K ∗L )∩(K ∗L ) = (K ∩K )∗(L ∩L ).
1 1 2 2 1 2 1 2
Lemma 3.3. Let K be a simplicial complex with two subcomplexes K and K . If K =
1 2
K ∪K , then Z (CX,X) = Z (CX,X)∪Z (CX,X) and Z (CX,X)∩Z (CX,X) =
1 2 K K1 K2 K1 K2
Z (CX,X).
K1∩K2
Lemma 3.4. Let K and L be simplicial complexes with disjoint index sets V and W,
respectively. Then,
Z (CX,X) = Z (CX ,X )×Z (CX ,X ),
K∗L K V V L W W
where X and X are the sub-sequences of X indexed by V and W, respectively.
V W
Proposition 3.5. Let K be a simplicial complex on [m] and X be a sequence of based CW-
complexes. If Z (CX,X) is a simply connected co-H-space, then Z (CX,X) ≃ Σ|K |∧
K K WI⊂[m] I
XI.
b
Proof. First, we remark that Σ|K |∧XI is also simply connected. By Theorem 1.2
WI⊂[m] I
b
we have that H ( Σ|K |∧XI) ∼= H (Z (CX,X)) = 0. Because Σ|K |∧XI is a
1 WI⊂[m] I 1 K WI⊂[m] I
b b
suspension, its fundamental group is a free group. Thus, π ( Σ|K |∧XI) = 0.
1 WI⊂[m] I
b
For a subset I ⊂ [m], the canonical projection p : CX → CX induces a map
I Qi∈[m] i Qi∈I i
Z (CX,X) → Z (CX ,X ), which is also denoted by p . We define a subset Z (CX ,X )′
K KI I I I KI I I
of Z (CX ,X ) by the equation
KI I I
Z (CX ,X )′ = {(x ,··· ,x ) ∈ Z (CX ,X ) | x = ∗ for some j},
KI I I i1 ik KI I I ij
A GOLOD COMPLEX WITH NON-SUSPENSION MOMENT-ANGLE COMPLEX 7
where I = {i ,··· ,i }. In [14], it is shown that Z (CX ,X )/Z (CX ,X )′ ≃ Σ|K |∧XI.
1 k KI I I KI I I I
b
Now, we consider the composite of maps
2m
f : ZK(CX,X) → _ZK(CX,X) −∨−I−⊂[−m−]p→I _ ZKI(CXI,XI)
I⊂[m]
→ _ ZKI(CXI,XI)/ZKI(CXI,XI)′ ≃ _ Σ|KI|∧XI,
b
I⊂[m] I⊂[m]
where the first map is the iterated co-multiplication of Z (CX,X). Because Z (CX,X) and
K K
Σ|K |∧XI are simply connected CW-complexes, to prove that f is homotopy equivalent
WI⊂[m] I
b
it suffices to show that f induces a homology isomorphism. In [15], it is shown that Σf is a
homotopy equivalence. In particular, f induces a homology isomorphism, andthus we complete
(cid:3)
the proof.
Proof of Theorem 2.3. ByLemmas3.1and3.2, wehavethat(K∗L)∨ = K∨∗∆W∪∆V∗L∨
and K∨ ∗∆V(L) ∩∆V(K) ∗L∨ = K∨ ∗L∨. Therefore, from Lemma 3.3 we obtain the following
push-out diagram of spaces:
Z (CX,X) −−−→ Z (CX,X)
K∨∗L∨ K∨∗∆W
 
 
y y
Z (CX,X) −−−→ Z (CX,X).
∆V∗L∨ (K∗L)∨
Here, we remark that K∨ ∗ L∨ and K∨ ∗ ∆W are non-void simplicial complexes, because we
assume that K 6= ∆V and L 6= ∆W. By Lemma 3.4, the above push-out diagram is equivalent
to the following push-out diagram:
Z (CX ,X )×Z (CX ,X ) −−−→ Z (CX ,X )× CX
K∨ V V L∨ W W K∨ V V Qw∈W w
  .
 
y y
CX ×Z (CX ,X ) −−−→ Z (CX,X).
Qv∈V v L∨ W W (K∗L)∨
Because CX and CX are contractible, the above diagram yields the following
Qv∈V v Qw∈W w
homotopy equivalences:
Z (CX,X) ≃ Z (CX ,X )∗Z (CX ,X )
(K∗L)∨ K∨ V V L∨ W W
≃ ΣZ (CX ,X )∧Z (CX ,X ).
K∨ V V L∨ W W
By Theorem 1.2, ΣZ (CX ,X ) is a double suspension, which implies that Z (CX,X)
K∨ V V (K∗L)∨
(cid:3)
is also a double suspension. By invoking Proposition 3.5, we complete the proof.
8 KOUYEMONIRIYEANDTATSUYAYANO
4. Proof of Theorem 2.5.
In this section, we prove Theorem 2.5. Again, we begin by stating some elementary lemmas.
Lemma 4.1. Let K and L be simplicial complexes without ghost vertices on index sets V
and W, respectively. Let α = V ∩ W be a common face of K and L. Then, (K ∪ L)∨ =
α
(∂∆V−α ∗∆α ∗∂∆W−α)∪(K∨ ∗∆W−α)∪(∆V−α ∗L∨).
Proof. For u ∈ V −α and v ∈ W −α, {u,v} is a minimal non-face of K∪ L, and a minimal
α
non-face of K or L is also a minimal non-face of K ∪ L. For any u ∈ V −α and v ∈ V −α,
α
V ∪W −{u,v} is a facet of (K ∪ L)∨. Furthermore, for any minimal non-face σ of K or L,
α
V ∪W −σ is a facet of (K ∪ L)∨. This implies the desired equality of the two complexes. (cid:3)
α
Lemma 4.2. If K is a simplicial complex with a ghost vertex, then the BBCG decomposition
for K∨ is desuspendable. In particular, K∨ is Golod.
Proof. Let K be a simplicial complex on [m] and v be a ghost vertex of K. Then, [m]−v is
a facet of K∨, and thus dimK∨ ≥ m−2. Then, it follows from Theorem 1.2 and Proposition
3.5 of [14] that the BBCG decomposition for K∨ is desuspendable. (cid:3)
Lemma 4.3. Let α be a face of a simplicial complex K on an index set V. If α is not a
facet of K, then the inclusion map
Z (CX,X) → Z (CX,X) = Z (CX,X)
K∨ (∆α)∨ ∂∆V−α∗∆α
is null homotopic.
Proof. Because α is not a facet of K, there is a face β of K such that α ( β. Then,
∆α ( ∆β ⊂ K, which implies that (∆α)∨ ) (∆β)∨ ⊃ K∨. That is, ∂∆V−α ∗ ∆α )
∂∆V−β ∗ ∆β ⊃ K∨. Therefore, the inclusion Z (CX,X) ֒→ Z (CX,X) factors
K∨ ∂∆V−α∗∆α
as Z (CX,X) → Z (CX,X) → Z (CX,X). To show that the inclusion
K∨ ∂∆V−β∗∆β ∂∆V−α∗∆α
Z (CX,X) → Z (CX,X) is null homotopic, it is sufficient to show that
K∨ ∂∆V−α∗∆α
Z (CX,X) → Z (CX,X) is null homotopic. Because
∂∆V−β∗∆β ∂∆V−α∗∆α
Z∂∆V−β∗∆β(CX,X) = Z∂∆V−β(CXV−β,XV−β)×YCXj,
j∈β
Z∂∆V−α∗∆α(CX,X) = Z∂∆V−α(CXV−α,XV−α)×YCXj,
j∈α
it suffices to show that the map f : Z (CX ,X ) → Z (CX ,X ) induced
∂∆V−β V−β V−β ∂∆V−α V−α V−α
bytheinclusionofsimplicialsets∂∆V−β → ∂∆V−α isnull-homotopic. Because∆V−β ⊂ ∂∆V−α,
the above map factors through a contractible space Z (CX ,X ), which implies that
∆V−β V−β V−β
f is null-homotopic, and thus we complete the proof. (cid:3)
In addition to the above, we require the following lemma to prove Theorem 2.5.
A GOLOD COMPLEX WITH NON-SUSPENSION MOMENT-ANGLE COMPLEX 9
Lemma 4.4 (Lemma 3.2 of [13]). Define Q as the push-out
ι×(1∨1)
A×(B ∨C) −−−−→ CA×(B ∨C)
1×(1∨∗) 
 
y y
A×(B ∨D) −−−→ Q,
where ι : A → CA is the inclusion. Then, the homotopy equivalence
≃
Q −→ B ∨Σ(A∧C)∨(A⋉D)
holds, which is natural with respect to A, B, C, and D, where X ⋉Y = (X ×Y)/(X ×∗).
Proof of Theorem 2.5. If K or L has a ghost vertex, then K∪ L also has a ghost vertex.
α
In this case, the BBCG decomposition is desuspendable, by Lemma 4.2. Therefore, we assume
that K and L do not have any ghost vertices. In the following proof, Z (CX,X) is abbreviated
K
as Z .
K
First, we will show that if K = ∆V and L 6= ∆W, then
Z ≃ (Z ⋉Z )∨Σ(Z ∧Z ).
(∆V∪αL)∨ ∂∆V−α ∂∆W−α ∂∆V−α L∨
It follows from Lemma 4.1 that (∆V ∪ L)∨ = (∂∆V−α∗∆α∗∂∆W−α)∪(∆V−α∗L∨). Then,
α
by Lemma 3.3 we have the push-out diagram of spaces
//
Z Z
∂∆V−α∗L∨ ∆V−α∗L∨
(cid:15)(cid:15) (cid:15)(cid:15)
//
Z Z ,
∂∆V−α∗∆α∗∂∆W−α (∆V∪αL)∨
which by Lemma 3.4 is equivalent to the following push-out diagram:
Z ×Z −−−→ Z ×Z
∂∆V−α L∨ ∆V−α L∨
id×incl 
 
y y
Z ×Z ×Z −−−→ Z .
∂∆V−α ∆α ∂∆W−α (∆V∪αL)∨
By Lemma 4.3, the inclusion map Z ֒→ Z × Z is null-homotopic. Therefore,
L∨ ∆α ∂∆W−α
Z is homotopy equivalent to the push-out P of the following diagram:
(∆V∪αL)∨
Z ×Z −−−→ Z ×Z
∂∆V−α L∨ ∆V−α L∨
(4.1) id×∗ 
 
y y
j
Z ×Z ×Z −−−→ P
∂∆V−α ∆α ∂∆W−α
By Lemma 4.4, P is homotopy equivalent to
(Z ⋉(Z ×Z ))∨Σ(Z ∧Z ) ≃ (Z ⋉Z )∨Σ(Z ∧Z ),
∂∆V−α ∆α ∂∆W−α ∂∆V−α L∨ ∂∆V−α ∂∆W−α ∂∆V−α L∨
10 KOUYEMONIRIYEANDTATSUYAYANO
and j in the diagram (4.1) can be identified with the following composite of canonical maps:
(4.2) Z ×Z ×Z → Z ⋉(Z ×Z )
∂∆V−α ∆α ∂∆W−α ∂∆V−α ∆α ∂∆W−α
֒→ (Z ⋉(Z ×Z ))∨Σ(Z ∧Z ).
∂∆V−α ∆α ∂∆W−α ∂∆V−α L∨
Thus, the following homotopy equivalence holds: Z ≃ (Z ⋉Z )∨Σ(Z ∧
(∆V∪αL)∨ ∂∆V−α ∂∆W−α ∂∆V−α
Z ). ThishomotopyequivalenceandExample2.2inducethefollowinghomotopyequivalences:
L∨
Z ≃ (Z ⋉Z )∨Σ(Z ∧Z )
(∆V∪αL)∨ ∂∆V−α ∂∆W−α ∂∆V−α L∨
≃ Z ∨(Z ∧Z )∨Σ(Z ∧Z ).
∂∆W−α ∂∆V−α ∂∆W−α ∂∆V−α L∨
By the above homotopy equivalence and the BBCG decomposition, it is easy to see that
Z ≃ Σ|((∆V ∪ L)∨) |XI. Here, we remark that we can apply Proposition
(∆V∪αL)∨ WI⊂[m] α I
b
3.5 if |W −α| ≥ 3.
Similarly, if K 6= ∆V and L = ∆W then we have Z ≃ (Z ⋊Z )∨Σ(Z ∧
(K∪α∆W)∨ ∂∆V−α ∂∆W−α K∨
Z ) ≃ Σ|((K ∪ ∆W)∨) |XI.
∂∆W−α WI⊂[m] α I
b
Next, we consider the case that K 6= ∆V and L 6= ∆W. Then, we have the push-out diagram
(∂∆V−α ∗∆α ∗∂∆W−α) −−−→ (∂∆V−α ∗∆α ∗∂∆W−α)∪(∆V−α ∗L∨)
 
 
y y
(∂∆V−α ∗∆α ∗∂∆W−α)∪(K∨ ∗∆W−α) −−−→ (K ∪ L)∨,
α
which induces the following push-out diagram of spaces, by Lemmas 3.3 and 3.4:
Z ×Z ×Z −−−→ Z
∂∆V−α ∆α ∂∆W−α (∂∆V−α∗∆α∗∂∆W−α)∪(∆V−α∗L∨)
  .
 
y y
Z −−−→ Z .
(∂∆V−α∗∆α∗∂∆W−α)∪(K∨∗∆W−α) (K∪αL)∨
Because Z × Z × Z → Z is a closed cofibration,
∂∆V−α ∆α ∂∆W−α (∂∆V−α∗∆α∗∂∆W−α)∪(∆V−α∗L∨)
Z is homotopy equivalent to the homotopy push-out of the following diagram:
(K∪αL)∨
Z ×Z ×Z −−−→ Z
∂∆V−α ∆α ∂∆W−α (∂∆V−α∗∆α∗∂∆W−α)∪(∆V−α∗L∨)
j1
j2
y
Z
(∂∆V−α∗∆α∗∂∆W−α)∪(K∨∗∆W−α)
Because
Z ≃ (Z ⋉(Z ×Z ))∨Σ(Z ∧Z )
(∂∆V−α∗∆α∗∂∆W−α)∪(∆V−α∗L∨) ∂∆V−α ∆α ∂∆W−α ∂∆V−α L∨
≃ (Z ⋉Z )∨Σ(Z ∧Z ),
∂∆V−α ∂∆W−α ∂∆V−α L∨
Z ≃ ((Z ×Z )⋊Z )∨Σ(Z ∧Z )
(∂∆V−α∗∆α∗∂∆W−α)∪(K∨∗∆W−α) ∂∆V−α ∆α ∂∆W−α K∨ ∂∆W−α
≃ (Z ⋊Z )∨Σ(Z ∧Z ),
∂∆V−α ∂∆W−α K∨ ∂∆W−α