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Decomposing real moment-angle complexes
Daisuke Kishimoto
with Kouyemon Iriye
Kyoto University
23 January 2014; Toric Topology in Osaka
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Our object
▶ Let K be a simplicial complex on the vertex set [m] = {1, . . . ,m}.▶ Let (X ,A) = {(Xi ,Ai )}i∈[m] be a sequence of pairs of spaces.
DefinitionThe polyhedral product ZK (X ,A) is defined as
ZK (X ,A) =∪σ∈K
D(σ) (⊂ X1 × · · · × Xm)
where D(σ) = Y1 × · · · × Ym for Yi =
{Xi i ∈ σ
Ai i ∈ σ.
Our object is the real moment-angle complex
ZK = ZK (D1, S0)
which is fundamental in studying the real version of quasitoricmanifolds (called small covers), right-angled Coxeter groups and etc.
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Our object
▶ Let K be a simplicial complex on the vertex set [m] = {1, . . . ,m}.▶ Let (X ,A) = {(Xi ,Ai )}i∈[m] be a sequence of pairs of spaces.
DefinitionThe polyhedral product ZK (X ,A) is defined as
ZK (X ,A) =∪σ∈K
D(σ) (⊂ X1 × · · · × Xm)
where D(σ) = Y1 × · · · × Ym for Yi =
{Xi i ∈ σ
Ai i ∈ σ.
Our object is the real moment-angle complex
ZK = ZK (D1, S0)
which is fundamental in studying the real version of quasitoricmanifolds (called small covers), right-angled Coxeter groups and etc.
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![Page 4: Daisuke Kishimoto with Kouyemon Iriyemasuda/toric2014_osaka/Kishimoto(slide).pdf · Decomposing real moment-angle complexes Daisuke Kishimoto with Kouyemon Iriye Kyoto University](https://reader031.vdocuments.site/reader031/viewer/2022022110/5c0dd83b09d3f20b788c0bd9/html5/thumbnails/4.jpg)
Motivation
Generalizing the decomposition
Σ(X × Y ) ≃ ΣX ∨ ΣY ∨ Σ(X ∧ Y ),
Bahri, Bendersky, Cohen & Gitler decomposed ΣZK (X ,A). As aspecial case, we have:
Theorem (Bahri, Bendersky, Cohen & Gitler ’10)
There is a homotopy equivalence
ΣZK (CX ,X ) ≃ Σ∨
∅=I⊂[m]
|ΣKI | ∧ X I
where KI is the maximum subcomplex of K on the vertex set I andX I =
∧i∈I Xi .
QuestionWhen does this decomposition desuspend?
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Motivation
Generalizing the decomposition
Σ(X × Y ) ≃ ΣX ∨ ΣY ∨ Σ(X ∧ Y ),
Bahri, Bendersky, Cohen & Gitler decomposed ΣZK (X ,A). As aspecial case, we have:
Theorem (Bahri, Bendersky, Cohen & Gitler ’10)
There is a homotopy equivalence
ΣZK (CX ,X ) ≃ Σ∨
∅=I⊂[m]
|ΣKI | ∧ X I
where KI is the maximum subcomplex of K on the vertex set I andX I =
∧i∈I Xi .
QuestionWhen does this decomposition desuspend?
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If the decomposition desuspends, ZK (CX ,X ) must be a suspension.So at least we need the following property for K .
DefinitionK is called Golod if all products in H∗(ZK (D
2, S1)) are trivial.
Many important simplicial complexes have the Golod property.
DefinitionK is Cohen-Macaulay (CM) if so is its Stanley-Reisner ring.
There is a homological characterization of CM complexes.
Theorem (Reisner ’76)
K is CM ⇐⇒ H∗(lkK (σ)) = 0 for ∗ < dim lkK (σ) and ∀σ ∈ K.
▶ The Alexander dual of K is defined as
K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is CM, K is Golod.
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If the decomposition desuspends, ZK (CX ,X ) must be a suspension.So at least we need the following property for K .
DefinitionK is called Golod if all products in H∗(ZK (D
2, S1)) are trivial.
Many important simplicial complexes have the Golod property.
DefinitionK is Cohen-Macaulay (CM) if so is its Stanley-Reisner ring.
There is a homological characterization of CM complexes.
Theorem (Reisner ’76)
K is CM ⇐⇒ H∗(lkK (σ)) = 0 for ∗ < dim lkK (σ) and ∀σ ∈ K.
▶ The Alexander dual of K is defined as
K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is CM, K is Golod.
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If the decomposition desuspends, ZK (CX ,X ) must be a suspension.So at least we need the following property for K .
DefinitionK is called Golod if all products in H∗(ZK (D
2, S1)) are trivial.
Many important simplicial complexes have the Golod property.
DefinitionK is Cohen-Macaulay (CM) if so is its Stanley-Reisner ring.
There is a homological characterization of CM complexes.
Theorem (Reisner ’76)
K is CM ⇐⇒ H∗(lkK (σ)) = 0 for ∗ < dim lkK (σ) and ∀σ ∈ K.
▶ The Alexander dual of K is defined as
K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is CM, K is Golod.
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If the decomposition desuspends, ZK (CX ,X ) must be a suspension.So at least we need the following property for K .
DefinitionK is called Golod if all products in H∗(ZK (D
2, S1)) are trivial.
Many important simplicial complexes have the Golod property.
DefinitionK is Cohen-Macaulay (CM) if so is its Stanley-Reisner ring.
There is a homological characterization of CM complexes.
Theorem (Reisner ’76)
K is CM ⇐⇒ H∗(lkK (σ)) = 0 for ∗ < dim lkK (σ) and ∀σ ∈ K.
▶ The Alexander dual of K is defined as
K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is CM, K is Golod.4 / 15
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CM complexes are pure, and sequentially Cohen-Macaulay (SCM)complexes are a nonpure generalization of CM complexes.
▶ Let K [d ] be the subcomplex of K generated by d-dim faces.
Theorem-Definition (Duval ’96)
K is SCM ⇐⇒ K [d ] is CM for ∀d ≥ 0.
Corollary
K is CM ⇐⇒ K is SCM and pure.
There is also a homological characterization of SCM complexes whichwe do not mention here.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is SCM, K is Golod.
QuestionDoes the decomposition of ΣZK (CX ,X ) desuspend if K∨ is SCM?
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CM complexes are pure, and sequentially Cohen-Macaulay (SCM)complexes are a nonpure generalization of CM complexes.
▶ Let K [d ] be the subcomplex of K generated by d-dim faces.
Theorem-Definition (Duval ’96)
K is SCM ⇐⇒ K [d ] is CM for ∀d ≥ 0.
Corollary
K is CM ⇐⇒ K is SCM and pure.
There is also a homological characterization of SCM complexes whichwe do not mention here.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is SCM, K is Golod.
QuestionDoes the decomposition of ΣZK (CX ,X ) desuspend if K∨ is SCM?
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CM complexes are pure, and sequentially Cohen-Macaulay (SCM)complexes are a nonpure generalization of CM complexes.
▶ Let K [d ] be the subcomplex of K generated by d-dim faces.
Theorem-Definition (Duval ’96)
K is SCM ⇐⇒ K [d ] is CM for ∀d ≥ 0.
Corollary
K is CM ⇐⇒ K is SCM and pure.
There is also a homological characterization of SCM complexes whichwe do not mention here.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is SCM, K is Golod.
QuestionDoes the decomposition of ΣZK (CX ,X ) desuspend if K∨ is SCM?
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CM complexes are pure, and sequentially Cohen-Macaulay (SCM)complexes are a nonpure generalization of CM complexes.
▶ Let K [d ] be the subcomplex of K generated by d-dim faces.
Theorem-Definition (Duval ’96)
K is SCM ⇐⇒ K [d ] is CM for ∀d ≥ 0.
Corollary
K is CM ⇐⇒ K is SCM and pure.
There is also a homological characterization of SCM complexes whichwe do not mention here.
Theorem (Herzog, Reiner & Welker ’99)
If K∨ is SCM, K is Golod.
QuestionDoes the decomposition of ΣZK (CX ,X ) desuspend if K∨ is SCM?
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There are implications of simplicial complexes:
shifted ⇒ vertex-decomposable ⇒ shellable ⇒ SCM
The desuspension of the decomposition of ΣZK (CX ,X ) was provedwhen K∨ is
▶ shifted by Grbic & Theriault ’12, Iriye & K ’12, and
▶ vertex-decomposable by Grujic & Welker ’13.
Finally we have:
Theorem (Iriye & K ’13)
If K∨ is SCM and each Xi is a connected finite CW-complex, then
ZK (CX ,X ) ≃∨
∅=I⊂[m]
|ΣKI | ∧ X I .
. . .but the theorem does not apply to real moment-angle complexes, forwhich I will explain a new method.
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There are implications of simplicial complexes:
shifted ⇒ vertex-decomposable ⇒ shellable ⇒ SCM
The desuspension of the decomposition of ΣZK (CX ,X ) was provedwhen K∨ is
▶ shifted by Grbic & Theriault ’12, Iriye & K ’12, and
▶ vertex-decomposable by Grujic & Welker ’13.
Finally we have:
Theorem (Iriye & K ’13)
If K∨ is SCM and each Xi is a connected finite CW-complex, then
ZK (CX ,X ) ≃∨
∅=I⊂[m]
|ΣKI | ∧ X I .
. . .but the theorem does not apply to real moment-angle complexes, forwhich I will explain a new method.
6 / 15
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There are implications of simplicial complexes:
shifted ⇒ vertex-decomposable ⇒ shellable ⇒ SCM
The desuspension of the decomposition of ΣZK (CX ,X ) was provedwhen K∨ is
▶ shifted by Grbic & Theriault ’12, Iriye & K ’12, and
▶ vertex-decomposable by Grujic & Welker ’13.
Finally we have:
Theorem (Iriye & K ’13)
If K∨ is SCM and each Xi is a connected finite CW-complex, then
ZK (CX ,X ) ≃∨
∅=I⊂[m]
|ΣKI | ∧ X I .
. . .but the theorem does not apply to real moment-angle complexes, forwhich I will explain a new method.
6 / 15
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Result
TheoremIf K∨ is SCM, then
ZK ≃∨
∅=I⊂[m]
|ΣKI |.
In showing the previous result, it was proved:
Lemma (Iriye & K ’13)
If K∨ is SCM, then for ∅ = ∀I ⊂ [m], |ΣKI | has the homotopy type ofa wedge of spheres.
Hence we get:
Corollary
If K∨ is SCM, ZK has the homotopy type of a wedge of spheres.
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Result
TheoremIf K∨ is SCM, then
ZK ≃∨
∅=I⊂[m]
|ΣKI |.
In showing the previous result, it was proved:
Lemma (Iriye & K ’13)
If K∨ is SCM, then for ∅ = ∀I ⊂ [m], |ΣKI | has the homotopy type ofa wedge of spheres.
Hence we get:
Corollary
If K∨ is SCM, ZK has the homotopy type of a wedge of spheres.
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StratificationTo prove the main theorem, we
▶ describe the stratification of ZK in terms of the cubical subdivisionof a simplicial complex in the book of Buchstaber and Panov, and
▶ show the triviality of strata when K∨ is SCM.
DefinitionFor i = 0, . . . ,m, we define
Z iK =
∪I⊂[m], |I |=i
ZKI
where ZKIlies in {(x1, . . . , xm) ∈ (D1)m | xj = −1 for j ∈ I}.
Then we get a stratification
∗ = Z 0K ⊂ Z 1
K ⊂ · · · ⊂ Zm−1K ⊂ Zm
K = ZK .
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StratificationTo prove the main theorem, we
▶ describe the stratification of ZK in terms of the cubical subdivisionof a simplicial complex in the book of Buchstaber and Panov, and
▶ show the triviality of strata when K∨ is SCM.
DefinitionFor i = 0, . . . ,m, we define
Z iK =
∪I⊂[m], |I |=i
ZKI
where ZKIlies in {(x1, . . . , xm) ∈ (D1)m | xj = −1 for j ∈ I}.
Then we get a stratification
∗ = Z 0K ⊂ Z 1
K ⊂ · · · ⊂ Zm−1K ⊂ Zm
K = ZK .
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Let us recall the cubical subdivision of K .
▶ For σ ⊂ τ ⊂ [m], put
Cσ⊂τ = {(x1, . . . , xm) ∈ (D1)m | xi = −1 (i ∈ σ),+1 (i ∈ τ)}
which is a (|τ | − |σ|)-dimensional face of (D1)m.
All faces of (D1)m not including (+1, . . . ,+1) are given by Cσ⊂τ .
▶ A piecewise linear map
ic : |Sd∆m−1| → (D1)m, σ 7→ Cσ⊂σ
is an embedding, where ∅ = σ ⊂ [m] are vertices of Sd∆m−1.
So ic(|Sd∆m−1|) is the union of all faces of (D1)m not including(+1, . . . ,+1), which is regarded as the cubical subdivision of ∆m−1.
▶ Define the embedding Cone(ic) : |Cone(Sd∆m−1)| → (D1)m asthe extension of ic which sends the cone point to (+1, . . . ,+1).
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Let us recall the cubical subdivision of K .
▶ For σ ⊂ τ ⊂ [m], put
Cσ⊂τ = {(x1, . . . , xm) ∈ (D1)m | xi = −1 (i ∈ σ),+1 (i ∈ τ)}
which is a (|τ | − |σ|)-dimensional face of (D1)m.
All faces of (D1)m not including (+1, . . . ,+1) are given by Cσ⊂τ .
▶ A piecewise linear map
ic : |Sd∆m−1| → (D1)m, σ 7→ Cσ⊂σ
is an embedding, where ∅ = σ ⊂ [m] are vertices of Sd∆m−1.
So ic(|Sd∆m−1|) is the union of all faces of (D1)m not including(+1, . . . ,+1), which is regarded as the cubical subdivision of ∆m−1.
▶ Define the embedding Cone(ic) : |Cone(Sd∆m−1)| → (D1)m asthe extension of ic which sends the cone point to (+1, . . . ,+1).
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Let us recall the cubical subdivision of K .
▶ For σ ⊂ τ ⊂ [m], put
Cσ⊂τ = {(x1, . . . , xm) ∈ (D1)m | xi = −1 (i ∈ σ),+1 (i ∈ τ)}
which is a (|τ | − |σ|)-dimensional face of (D1)m.
All faces of (D1)m not including (+1, . . . ,+1) are given by Cσ⊂τ .
▶ A piecewise linear map
ic : |Sd∆m−1| → (D1)m, σ 7→ Cσ⊂σ
is an embedding, where ∅ = σ ⊂ [m] are vertices of Sd∆m−1.
So ic(|Sd∆m−1|) is the union of all faces of (D1)m not including(+1, . . . ,+1), which is regarded as the cubical subdivision of ∆m−1.
▶ Define the embedding Cone(ic) : |Cone(Sd∆m−1)| → (D1)m asthe extension of ic which sends the cone point to (+1, . . . ,+1).
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SSSSSS
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Figure : The embedding ic : |Sd∆2| → (D1)3
▶ Define the embeddings
ic : |SdK | → (D1)m, Cone(ic) : |Cone(SdK )| → (D1)m
as the restriction of the above embeddings.
These are regarded as the cubical subdivisions of K and its cone.
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Figure : The embedding ic : |Sd∆2| → (D1)3
▶ Define the embeddings
ic : |SdK | → (D1)m, Cone(ic) : |Cone(SdK )| → (D1)m
as the restriction of the above embeddings.
These are regarded as the cubical subdivisions of K and its cone.
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By definition, we have
ZmK =
∪σ⊂τ⊂[m]τ−σ∈K
Cσ⊂τ , Zm−1K =
∪∅=σ⊂τ⊂[m]
τ−σ∈K
Cσ⊂τ
and
Cone(ic)(|Cone(SdK )|) =∪
σ⊂τ∈KCσ⊂τ , ic(|SdK |) =
∪∅=σ⊂τ∈K
Cσ⊂τ .
Then the map Cone(ic) : |Cone(SdK )| → (D1)m descends to
Cone(ic) : (|Cone(SdK )|, |SdK |) → (ZmK ,Zm−1
K ).
On the other hand, since
ZmK − Zm−1
K =∪
σ⊂τ∈KCσ⊂τ −
∪∅=σ⊂τ∈K
Cσ⊂τ
= Cone(ic)(|Cone(SdK )|)− ic(|SdK |),
the above map is a relative homeomorphism.
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By definition, we have
ZmK =
∪σ⊂τ⊂[m]τ−σ∈K
Cσ⊂τ , Zm−1K =
∪∅=σ⊂τ⊂[m]
τ−σ∈K
Cσ⊂τ
and
Cone(ic)(|Cone(SdK )|) =∪
σ⊂τ∈KCσ⊂τ , ic(|SdK |) =
∪∅=σ⊂τ∈K
Cσ⊂τ .
Then the map Cone(ic) : |Cone(SdK )| → (D1)m descends to
Cone(ic) : (|Cone(SdK )|, |SdK |) → (ZmK ,Zm−1
K ).
On the other hand, since
ZmK − Zm−1
K =∪
σ⊂τ∈KCσ⊂τ −
∪∅=σ⊂τ∈K
Cσ⊂τ
= Cone(ic)(|Cone(SdK )|)− ic(|SdK |),
the above map is a relative homeomorphism.
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By definition, we have
ZmK =
∪σ⊂τ⊂[m]τ−σ∈K
Cσ⊂τ , Zm−1K =
∪∅=σ⊂τ⊂[m]
τ−σ∈K
Cσ⊂τ
and
Cone(ic)(|Cone(SdK )|) =∪
σ⊂τ∈KCσ⊂τ , ic(|SdK |) =
∪∅=σ⊂τ∈K
Cσ⊂τ .
Then the map Cone(ic) : |Cone(SdK )| → (D1)m descends to
Cone(ic) : (|Cone(SdK )|, |SdK |) → (ZmK ,Zm−1
K ).
On the other hand, since
ZmK − Zm−1
K =∪
σ⊂τ∈KCσ⊂τ −
∪∅=σ⊂τ∈K
Cσ⊂τ
= Cone(ic)(|Cone(SdK )|)− ic(|SdK |),
the above map is a relative homeomorphism.11 / 15
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More generally, we have:
Proposition
The map
Cone(ic) :⨿
I⊂[m], |I |=i
(|Cone(SdKI )|, |SdKI |) → (Z iK ,Z
i−1K )
is a relative homoemorphism.
Corollary
Z iK is obtained from Z i−1
K by attaching cones to ic(|SdKI |) ⊂ Z i−1K for
∀I ⊂ [m] with |I | = i .
Corollary
If ic : |SdKI | → Z|I |−1K is null homotopic for all ∅ = I ⊂ [m], then
ZK ≃∨
∅=I⊂[m]
|ΣKI |.
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More generally, we have:
Proposition
The map
Cone(ic) :⨿
I⊂[m], |I |=i
(|Cone(SdKI )|, |SdKI |) → (Z iK ,Z
i−1K )
is a relative homoemorphism.
Corollary
Z iK is obtained from Z i−1
K by attaching cones to ic(|SdKI |) ⊂ Z i−1K for
∀I ⊂ [m] with |I | = i .
Corollary
If ic : |SdKI | → Z|I |−1K is null homotopic for all ∅ = I ⊂ [m], then
ZK ≃∨
∅=I⊂[m]
|ΣKI |.
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SCM case
LemmaIf K∨ is SCM, so is (KI )
∨ for any ∅ = I ⊂ [m].
Then it is sufficient to show that ic : |SdK | → Zm−1K is null homotopic.
▶ Let K be the simplicial complex obtained from K by adding allmissing faces.
LemmaThe map ic : |SdK | → Zm−1
K factors as
|SdK | incl−−→ |SdK | → Zm−1K .
Proposition
If K∨ is SCM, then for each prime p, there is a simplicial complex ∆such that
K ⊂ ∆ ⊂ K and |∆|(p) ≃ ∗.
In particular, the map |SdK | incl−−→ |SdK | is null homotopic.
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SCM case
LemmaIf K∨ is SCM, so is (KI )
∨ for any ∅ = I ⊂ [m].
Then it is sufficient to show that ic : |SdK | → Zm−1K is null homotopic.
▶ Let K be the simplicial complex obtained from K by adding allmissing faces.
LemmaThe map ic : |SdK | → Zm−1
K factors as
|SdK | incl−−→ |SdK | → Zm−1K .
Proposition
If K∨ is SCM, then for each prime p, there is a simplicial complex ∆such that
K ⊂ ∆ ⊂ K and |∆|(p) ≃ ∗.
In particular, the map |SdK | incl−−→ |SdK | is null homotopic.
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SCM case
LemmaIf K∨ is SCM, so is (KI )
∨ for any ∅ = I ⊂ [m].
Then it is sufficient to show that ic : |SdK | → Zm−1K is null homotopic.
▶ Let K be the simplicial complex obtained from K by adding allmissing faces.
LemmaThe map ic : |SdK | → Zm−1
K factors as
|SdK | incl−−→ |SdK | → Zm−1K .
Proposition
If K∨ is SCM, then for each prime p, there is a simplicial complex ∆such that
K ⊂ ∆ ⊂ K and |∆|(p) ≃ ∗.
In particular, the map |SdK | incl−−→ |SdK | is null homotopic.13 / 15
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Generalization
Define Z iK (CX ,X ) ⊂ ZK (CX ,X ) similarly to Z i
K ⊂ ZK . Then there isa stratification
∗ = Z 0K (CX ,X ) ⊂ Z 1
K (CX ,X ) ⊂ · · · ⊂ ZmK (CX ,X ) = ZK (CX ,X ).
The composite
|Cone(SdK )| × X1 × · · · × Xmic×1−→ (D1)m × X1 × · · · × Xm
perm−→ (D1 × X1)× · · · × (D1 × Xm)
proj−→ CX1 × · · · × CXm
descends to a relative homeomorphism
(|Cone(SdK )|, |SdK )|)× (X ,F ) → (ZmK (CX ,X ),Zm−1
K (CX ,X ))
where X = X1 × · · · × Xm and F is the fat wedge of X1, . . . ,Xm.
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Generalization
Define Z iK (CX ,X ) ⊂ ZK (CX ,X ) similarly to Z i
K ⊂ ZK . Then there isa stratification
∗ = Z 0K (CX ,X ) ⊂ Z 1
K (CX ,X ) ⊂ · · · ⊂ ZmK (CX ,X ) = ZK (CX ,X ).
The composite
|Cone(SdK )| × X1 × · · · × Xmic×1−→ (D1)m × X1 × · · · × Xm
perm−→ (D1 × X1)× · · · × (D1 × Xm)
proj−→ CX1 × · · · × CXm
descends to a relative homeomorphism
(|Cone(SdK )|, |SdK )|)× (X ,F ) → (ZmK (CX ,X ),Zm−1
K (CX ,X ))
where X = X1 × · · · × Xm and F is the fat wedge of X1, . . . ,Xm.
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We can get an analogous relative homemorphism for the pair
(Z iK (CX ,X ),Z i−1
K (CX ,X ))
(i = 1, . . . ,m). Then we obtain that Z iK (CX ,X ) is constructed from
Z i−1K (CX ,X ) by attaching certain spaces, where the attaching maps
are explicitly described as above.
Using the previous result of Iriye & K, we can prove that the attachingmaps are null homotopic when K∨ is SCM. Therefore we obtain:
TheoremIf K∨ is SCM and each Xi is a CW-complex whose components arefinite complexes, then
ZK (CX ,X ) ≃∨
∅=I⊂[m]
|ΣKI | ∧ X I .
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We can get an analogous relative homemorphism for the pair
(Z iK (CX ,X ),Z i−1
K (CX ,X ))
(i = 1, . . . ,m). Then we obtain that Z iK (CX ,X ) is constructed from
Z i−1K (CX ,X ) by attaching certain spaces, where the attaching maps
are explicitly described as above.
Using the previous result of Iriye & K, we can prove that the attachingmaps are null homotopic when K∨ is SCM. Therefore we obtain:
TheoremIf K∨ is SCM and each Xi is a CW-complex whose components arefinite complexes, then
ZK (CX ,X ) ≃∨
∅=I⊂[m]
|ΣKI | ∧ X I .
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