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§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
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
—Some references
§1 Background—A triangle
Combinatorics
—Abstract simplicial
—Stanley-Reisnerface rings complexes
complexes
Algebra Topology
—Moment-angle
𝑎 𝑏
𝑐
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
—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.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
—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.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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}.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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}.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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—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].
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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—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 ).
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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—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
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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
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.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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
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 .
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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
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 .
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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
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
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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
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}.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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
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.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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]}.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes
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)
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes
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 ).
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes
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 ))∣
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes
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] ∖ �.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes
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 .
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§3.1 An algebra-combinatorics formula§3.2 Cohomology of a class of generalized moment-angle complexes
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 .
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§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).
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§4.1 Halperin-Carlsson conjecture§4.2 Our result
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.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§4.1 Halperin-Carlsson conjecture§4.2 Our result
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∣.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§4.1 Halperin-Carlsson conjecture§4.2 Our result
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).
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
§1 Background—A triangle§2 Notions and known results
§3 Further development§4 Application to Halperin-Carlsson conjecture
§4.1 Halperin-Carlsson conjecture§4.2 Our result
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.
Zhi Lu Mobius transform, moment-angle complexes and Halperin-Carlsson conjecture —A joint work with Xiangyu Cao
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