Daisuke Kishimoto with Kouyemon Iriyemasuda/toric2014_osaka/Kishimoto(slide).pdf · Decomposing...

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.

2 / 15

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.

2 / 15

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?

3 / 15

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?

3 / 15

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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K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.

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K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.

4 / 15

K∨ = {σ ⊂ [m] | [m]− σ ∈ K}.

If K∨ is CM, K is Golod.4 / 15

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.

If K∨ is SCM, K is Golod.

QuestionDoes the decomposition of ΣZK (CX ,X ) desuspend if K∨ is SCM?

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Corollary

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Corollary

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Corollary

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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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Finally we have:

ZK (CX ,X ) ≃∨

∅=I⊂[m]

|ΣKI | ∧ X I .

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Finally we have:

ZK (CX ,X ) ≃∨

∅=I⊂[m]

|ΣKI | ∧ X I .

6 / 15

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.

7 / 15

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

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 .

8 / 15

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

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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Cσ⊂τ = {(x1, . . . , xm) ∈ (D1)m | xi = −1 (i ∈ σ),+1 (i ∈ τ)}

ic : |Sd∆m−1| → (D1)m, σ 7→ Cσ⊂σ

9 / 15

Cσ⊂τ = {(x1, . . . , xm) ∈ (D1)m | xi = −1 (i ∈ σ),+1 (i ∈ τ)}

ic : |Sd∆m−1| → (D1)m, σ 7→ Cσ⊂σ

9 / 15

SSSSSS

��

�� QQss

{1} {2}

{1, 2}

{1, 3} {2, 3}{1, 2, 3} -ic

��

��s

s(−1, 1,−1)

(−1, 1, 1)

(−1,−1, 1)

(−1,−1,−1)

(1,−1, 1)

(1,−1,−1)

(1, 1,−1)

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.

10 / 15

SSSSSS

��

�� QQss

{1} {2}

{1, 2}

{1, 3} {2, 3}{1, 2, 3} -ic

��

��s

s(−1, 1,−1)

(−1, 1, 1)

(−1,−1, 1)

(−1,−1,−1)

(1,−1, 1)

(1,−1,−1)

(1, 1,−1)

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.

10 / 15

By definition, we have

∪σ⊂τ⊂[m]τ−σ∈K

Cσ⊂τ , Zm−1K =

∪∅=σ⊂τ⊂[m]

τ−σ∈K

Cσ⊂τ

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

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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∪∅=σ⊂τ⊂[m]

τ−σ∈K

Cσ⊂τ

∪∅=σ⊂τ∈K

Cσ⊂τ .

ZmK − Zm−1

K =∪

∪∅=σ⊂τ∈K

Cσ⊂τ

the above map is a relative homeomorphism.

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∪∅=σ⊂τ⊂[m]

τ−σ∈K

Cσ⊂τ

∪∅=σ⊂τ∈K

Cσ⊂τ .

ZmK − Zm−1

K =∪

∪∅=σ⊂τ∈K

Cσ⊂τ

the above map is a relative homeomorphism.11 / 15

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 |.

12 / 15

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 |.

12 / 15

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

∨ for any ∅ = I ⊂ [m].

K factors as

Proposition

K ⊂ ∆ ⊂ K and |∆|(p) ≃ ∗.

In particular, the map |SdK | incl−−→ |SdK | is null homotopic.

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SCM case

∨ for any ∅ = I ⊂ [m].

K factors as

Proposition

K ⊂ ∆ ⊂ K and |∆|(p) ≃ ∗.

In particular, the map |SdK | incl−−→ |SdK | is null homotopic.13 / 15

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 .

15 / 15

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 .

15 / 15

Daisuke Kishimoto with Kouyemon Iriyemasuda/toric2014_osaka/Kishimoto(slide).pdf · Decomposing...

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