m obius transform, moment-angle complexes and halperin ...masuda/transformation36_2009/lu.pdf ·...

$: M obius transform, moment-angle complexes and Halperin ...masuda/transformation36_2009/Lu.pdf · Let [m] = f1;:::;mg. {Abstract simplicial complexes on [m] An abstract simplicial$
§1 Background—A triangle§2 Notions and known results

§3 Further development§4 Application to Halperin-Carlsson conjecture

Mobius transform, moment-angle complexes andHalperin-Carlsson conjecture—A joint work with Xiangyu Cao

Zhi Lu

School of Mathematical ScienceFudan University, Shanghai

December 12, 2009

Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao



—Some references

§1 Background—A triangle

Combinatorics

—Abstract simplicial

—Stanley-Reisnerface rings complexes

complexes

Algebra Topology

—Moment-angle

𝑎 𝑏

𝑐




—Some references

ReferencesFor the edge a, see[1] Stanley, Richard P, Combinatorics and commutativealgebra. Second edition, Progress in Mathematics, 41,Birkhauser Boston, Inc., Boston, MA, 1996.[2] E. Miller and B. Sturmfels, Combinatorial CommutativeAlgebra, Graduate Texts in Math. 227, Springer, 2005.

For other two edges b, c , see[3] V. M. Buchstaber and T. E. Panov, Torus actions and theirapplications in topology and combinatorics, University LectureSeries, Vol. 24, Amer. Math. Soc., Providence, RI, 2002.




§2.1 Notion—Abstract simplicial complex§2.1 Notion—Stanley-Reisner face ring and Tor-algebra§2.1 Notion—Moment-angle complex§2.2 Known result on edge a—Hochster Theorem§2.2 Known result on edge c—Buchstaber-Panov Theorem

Notion—Abstract simplicial complex

Let [m] = {1, ...,m}.

–Abstract simplicial complexes on [m]

An abstract simplicial complex K on [m] is a collection ofsome subsets in [m] such that for each a ∈ K , any subset(including ∅) of a belongs to K .

Each a in K is called a simplex of dim= ∣a∣ − 1, anddimK = maxa∈K{dim a}.





Notion—Stanley-Reisner face ring

K : an abstract simplicial complex on [m]k: a field.

Stanley-Reisner face ring

k(K ) = k[v1, ..., vm]/IK

is called the Stanley-Reisner face ring of K , and IK is the idealgenerated by all square-free monomials vi1 ⋅ ⋅ ⋅ vis with� = {i1, ..., is} ∕∈ K .

RK: write k[v] = k[v1, ..., vm].





Notion—Betti numbers of Stanley-Reisner face ring k(K )

It is well-known that k(K ) is a finitely generated ℕm-gradedk[v]-module and it has an minimal free resolution

0←− k(K )←− F0�1←− F1 ←− ⋅ ⋅ ⋅ ←− Fh−1

�h←− Fh ←− 0 (1)

Write Fi =⊕

a∈ℕm

(k[v](−a)⊕ ⋅ ⋅ ⋅ ⊕ k[v](−a)︸︷︷︸

�k(K)i,a

)where k[v](−a) is

the ideal ⟨va⟩, and va = va11 ⋅ ⋅ ⋅ vamm for a = (a1, ..., am) ∈ ℕm.

Betti number

�k(K)i ,a ∈ ℕ is called the (i , a)-th Betti number of k(K ).





Notion—Tor-algebra of Stanley-Reisner face ring k(K )

Applying the functor ⊗k[v]k to the sequence (1) above, one mayobtain the following chain complex of ℕm-graded k[v]-modules:

0←− F0 ⊗k[v] k�′1←− F1 ⊗k[v] k←− ⋅ ⋅ ⋅

�′h←− Fh ⊗k[v] k←− 0.

Define Tork[v]i (k(K ), k) :=

ker �′iIm�′i+1

= Fi ⊗k[v] k so

dimk Tork[v]i (k(K ), k) = rankFi =

∑a∈ℕm

�k(K)i ,a .

Tor-algebra

Tork[v](k(K ), k) =h⊕

i=0

Tork[v]i (k(K ), k) =

⊕i∈[0,h]∩ℕa∈ℕm

Tork[v]i (k(K ), k)a





A remark

It is well-known that if a ∈ ℕm is not a vector in {0, 1}m, then

Tork[v]i (k(K ), k)a = 0, so �

k(K)i ,a = 0.

{0, 1}m ←→ 2m

⇓

write�k(K)i ,a := �

k(K)i ,a

where 2[m] ∋ a←→ a ∈ {0, 1}m.





Moment-angle complex

A general construction

K : a simplicial complex on vertex set [m] = {1, ...,m}(X ,W ): a pair of top. spaces with W ⊂ X .

K (X ,W ) :=∪�∈K

(∏i∈�

X ×∏i ∕∈�

W ) ⊆ Xm.

ZK := K (D2, S1) ⊂ (D2)m is called the moment-anglecomplex on K .

ℝZK := K (D1, S0) ⊂ (D1)m is called the real moment-anglecomplex on K .





Actions on ZK and ℝZK

A canonical action on ZK

D2 ={z ∈ ℂ

∣∣∣z ∣ ≤ 1}

and S1 = ∂D2.Since (D2)m ⊂ ℂm is invariant under the standard action of Tm

on ℂm given by((g1, ..., gm), (z1, ..., zm)

)7−→ (g1z1, ..., gmzm),

(D2)m admits a natural Tm-action whose orbit space is the unitcube Im ⊂ ℝm

≥0. The action Tm ↷ (D2)m then induces acanonical Tm-action Φ on ZK .

Similarly

A canonical action on ℝZK

ℝZK admits a canonical (ℤ2)m-action Φℝ on ℝZK





Hochster Theorem

On the edge a of the triangle, there is the following essential result:

Hochster Theorem

For each a ∈ 2[m],

H ∣a∣−i−1(K ∣a; k) ∼= Tork[v]i (k(K ), k)a

where K ∣a = {� ∈ K∣∣� ⊆ a}.





Buchstaber-Panov Theorem

On the edge c of the triangle, there is the following essential result:

Buchstaber-Panov Theorem

As k-algebras,

H∗(ZK ; k) ∼= Tork[v](k(K ), k)

where k(K ) = k[v]/IK = k[v1, ..., vm]/IK with deg vi = 2, and k isa field.




§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes

Further development—A viewpoint of analysis

Let 2[m]∗ ={f∣∣f : 2[m] −→ ℤ/2ℤ = {0, 1}

}. 2[m]∗ forms an

algebra over ℤ/2ℤ in the usual way, and it has a naturalbasis {�a∣a ∈ 2[m]} where �a is defined as follows:�a(b) = 1⇐⇒ b = a.

Given a f ∈ 2[m]∗, set

supp(f ) := f −1(1)

f is said to be nice if supp(f ) is an abstract simplicialcomplex.

A one-one correspondence

{all nice functions in 2[m]∗} ←→ { all abst. sim. subcpxes in 2[m]}.





Further development—An algebra-combinatorics formula

Mobius transform

On 2[m]∗, define a ℤ/2ℤ-valued Mobius transform

ℳ : 2[m]∗ −→ 2[m]∗

by the following way: for any f ∈ 2[m]∗ and a ∈ 2[m],

ℳ(f )(a) =∑b⊆a

f (b)





Further development—An algebra-combinatorics formula

The following result indicates an essential relationship betweenℳ(f ) and the Betti numbers of k(Kf ).

Algebra–combinatorics formula (Cao-Lu)

Suppose that f ∈ 2[m]∗ is nice such that Kf = supp(f ) is anabstract simplicial complex on [m]. Then

ℳ(f ) =h∑

i=0

∑a∈2[m]

�k(Kf )i ,a �a

where h denotes the length of the minimal free resolution of k(Kf ),

and �k(Kf )i ,a ’s denote the Betti numbers of k(Kf ).





An algebra-combinatorics formula

Corollary

∣supp(ℳ(f ))∣ ≤h∑

i=0

∑a∈2[m]

�k(Kf )i ,a .

Proof.ℳ(f ) =

∑hi=0

∑a∈2[m] �

k(Kf )i ,a �a =

∑a∈2[m]

(∑hi=0 �

k(Kf )i ,a

)�a

=⇒ for any a ∈ supp(ℳ(f )),∑h

i=0 �k(Kf )i ,a must be odd so∑h

i=0 �k(Kf )i ,a ≥ 1. Therefore

h∑i=0

∑a∈2[m]

�k(Kf )i ,a ≥

∑a∈supp(ℳ(f ))

h∑i=0

�k(Kf )i ,a ≥

∑a∈supp(ℳ(f ))

1 = ∣supp(ℳ(f ))∣





Generalized moment-angle complex

Given an abstract simplicial complex K on [m], let(X ,W ) = {(Xi ,Wi )}mi=1 be m pairs of CW-complexes withWi ⊂ Xi . Then the generalized moment-angle complex is definedas follows:

K (X ,W ) =∪�∈K

B�(X ,W ) ⊂m∏i=1

Xi

where B�(X ,W ) =∏m

i=1Hi and Hi =

{Xi if i ∈ �Wi if i ∈ [m] ∖ �.





A class of generalized moment-angle complexes

Take (X ,W ) = (D, S) = {(Di ,Si )}mi=1 with each CW-complex pair(Di , Si ) subject to the following conditions:

(1) Di is acyclic, that is, Hj(Di ) = 0 for any j .

(2) There exists a unique �i such that H�i (Si ) = ℤ and

Hj(Si ) = 0 for any j ∕= �i .

Then our objective is to calculate the cohomology of

Z(D,S)K := K (D,S) =

∪�∈K

B�(D, S) ⊂m∏i=1

Di .





Further development—Cohomology of a class ofgeneralized moment-angle complexes

Theorem (Cao-Lu)

As graded k-modules,

H∗(Z(D,S)K ; k) ∼= Tork[v](k(K ), k).

Corollary ∑i

dimkHi (Z(D,S)

K ; k) =h∑

i=0

∑a∈2[m]

�k(K)i ,a .




§4.1 Halperin-Carlsson conjecture§4.2 Our result

Application to Halperin-Carlsson conjecture

Halperin-Carlsson conjectureIf a finite-dimensional paracompact Hausdorff space X admits afree action of a torus T r (resp. a p-torus (ℤp)r , p prime) of rank r ,then the total dimension of its cohomology,∑

i

dimkHi (X ; k) ≥ 2r

where k is a field of characteristic 0 (resp. p).





Remark

Historically, the above conjecture in the p-torus caseoriginates from the work of P. A. Smith in 1950s.

For the case of a p-torus (ℤp)r freely acting on a finiteCW-complex homotopic to (Sn)k suggested by P. E. Conner,the problem has made an essential progress.

In the general case, the inequality was conjectured by S.Halperin for the torus case, and by G. Carlsson for the p-toruscase.

So far, the conjecture holds if r ≤ 3 in the torus and 2-toruscases and if r ≤ 2 in the odd p-torus case. Also, manyauthors have given contributions to the conjecture in manydifferent aspects.





Lower bound

Recall that

∑i

dimkHi (Z(D,S)

Kf; k) =

h∑i=0

∑a∈2[m]

�k(Kf )i ,a ≥ ∣supp(ℳ(f ))∣.

We can upbuild a method of compressing supp(f ) to get thedesired lower bound of ∣supp(ℳ(f ))∣.

Theorem (Cao-Lu)

For any nice f ∈ 2[m]∗, there exists some a ∈ supp(f ) such that

∣supp(ℳ(f ))∣ ≥ 2m−∣a∣.





Application to free actions

Theorem (Cao-Lu)

Let H (resp. Hℝ) be a rank r subtorus of Tm (resp. (ℤ2)m). If H(resp. Hℝ) can act freely on ZK (resp. ℝZK ), then∑

i

dimkHi (ZK ; k) =

∑i

dimkHi (ℝZK ; k) ≥ 2r .

Remark

The action of H (resp. Hℝ) on ZK (resp. ℝZK ) is naturallyregarded as the restriction of the Tm-action Φ to H (resp. the(ℤ2)m-action Φℝ to Hℝ).





Application to free actions

Corollary

The Halperin–Carlsson conjecture holds for ZK (resp. ℝZK ) underthe restriction of the Tm-action Φ (resp. the (ℤ2)m-action Φℝ).

Remark

Using a different method, Yury Ustinovsky has also recently provedthe Halperin’s toral rank conjecture for the moment-anglecomplexes with the restriction of natural tori actions, seearXiv:0909.1053.


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