Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture
aa r X i v : . [ m a t h . C O ] S e p M ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES ANDHALPERIN–CARLSSON CONJECTURE
XIANGYU CAO AND ZHI L ¨U
Abstract.
In this paper, we give an algebra–combinatorics formula of theM¨obius transform for an abstract simplicial complex K on [ m ] = { , ..., m } interms of the Betti numbers of the Stanley–Reisner face ring of K . Furthermore,we employ a way of compressing K to estimate the lower bound of the sum ofthose Betti numbers by using this formula. As an application, associating withthe moment-angle complex Z K (resp. real moment-angle complex R Z K ) of K , we show that the Halperin–Carlsson conjecture holds for Z K (resp. R Z K )under the restriction of the natural T m -action on Z K (resp. ( Z ) m -action on R Z K ). Introduction
Throughout this paper, assume that m is a positive integer and [ m ] = { , ..., m } .Also, k ℓ denotes the field of characteristic ℓ and k denotes a field of arbitrarycharacteristic. Let 2 [ m ] ∗ = (cid:8) f (cid:12)(cid:12) f : 2 [ m ] −→ Z / Z = { , } (cid:9) consisting of all Z / Z -valued functions on the power set 2 [ m ] . 2 [ m ] ∗ forms an algebraover Z / Z in the usual way, and it has a natural basis { δ a | a ∈ [ m ] } where δ a isdefined as follows: δ a ( b ) = 1 ⇐⇒ b = a . Given a f ∈ [ m ] ∗ , the inverse imageof f at 1 is called the support of f , denoted by supp( f ). f is said to be nice ifsupp( f ) is an abstract simplicial complex. Thus, we can identify all nice functionsin 2 [ m ] ∗ with all abstract simplicial subcomplexes in 2 [ m ] . On 2 [ m ] ∗ , we then definea Z / Z -valued M¨obius transform M : 2 [ m ] ∗ −→ [ m ] ∗ by the following way: for any f ∈ [ m ] ∗ and a ∈ [ m ] , M ( f )( a ) = P b ⊆ a f ( b ).Now let f ∈ [ m ] ∗ be nice such that K f = supp( f ) is an abstract simplicial com-plex on [ m ], and let k ( K f ) be the Stanley–Reisner face ring of K f . The followingresult indicates an essential relationship between M ( f ) and the Betti numbers of k ( K f ). Theorem 1.1 (Algebra–combinatorics formula) . Suppose that f ∈ [ m ] ∗ is nicesuch that K f = supp( f ) is an abstract simplicial complex on [ m ] . Then M ( f ) = h X i =0 X a ∈ [ m ] β k ( K f ) i,a δ a Mathematics Subject Classification.
Primary 05E45, 05E40, 13F55, 57S25; Secondary55U05, 16D03, 18G15, 57S10.
Key words and phrases.
M¨obius transform, moment-angle complex, Halperin–Carlssonconjecture.Supported by grants from FDUROP (No. 080705) and NSFC (No. 10671034, No. J0730103). where h denotes the length of the minimal free resolution of k ( K f ) , and β k ( K f ) i,a ’sdenote the Betti numbers of k ( K f ) ( see Definition 2.4 ) . The formula of Theorem 1.1 leads to the following inequality | supp( M ( f )) | ≤ h X i =0 X a ∈ [ m ] β k ( K f ) i,a . See Corollary 3.2. Then we use an approach of compressing supp( f ) to furtheranalyze the lower bound of | supp( M ( f )) | , and the result is stated as follows. Theorem 1.2.
Let f ∈ [ m ] ∗ be nice such that K f = supp( f ) is an abstractsimplicial complex on [ m ] . Then there exists some a ∈ supp( f ) such that | supp( M ( f )) | ≥ m −| a | . Remark . Since a ∈ supp( f ), | a | ≤ dim K f + 1, so | supp( M ( f )) | ≥ m −| a | ≥ m − dim K f − .As a result, we can consider the Halperin–Carlsson conjecture in the category of(real) moment-angle complexes. Let Z K (resp. R Z K ) be the moment-angle complex(resp. real moment-angle complex) on K where K is an abstract simplicial complexon vertex set [ m ]. Then we have that P i dim k H i ( Z K ; k ) = P i dim k H i ( R Z K ; k )for any k (see Theorem 4.2). Z K (resp. R Z K ) naturally admits a T m -action Φ(resp. ( Z ) m -action Φ R ). Theorem 1.3.
Let H ( resp. H R ) be a rank r subtorus of T m ( resp. ( Z ) m ) . If H ( resp. H R ) can act freely on Z K ( resp. R Z K ) , then X i dim k H i ( Z K ; k ) = X i dim k H i ( R Z K ; k ) ≥ r . Remark . In Theorem 1.3, the action of H (resp. H R ) on Z K (resp. R Z K ) isnaturally regarded as the restriction of the T m -action Φ to H (resp. the ( Z ) m -action Φ R to H R ). Corollary 1.4.
The Halperin–Carlsson conjecture holds for Z K ( resp. R Z K ) underthe restriction of the T m -action Φ ( resp. the ( Z ) m -action Φ R ) . † Remark . Following [16], the Halperin–Carlsson conjecture is stated as follows: • Let X be a finite-dimensional paracompact Hausdorff space. If X admitsa free action of a torus T r (resp. a p -torus ( Z p ) r , p prime) of rank r , then(1.1) X i dim k ℓ H i ( X ; k ℓ ) ≥ r where ℓ is 0 (resp. p ).Historically, the above conjecture in the p -torus case originates from the work of P.A. Smith ([17]). For the case of a p -torus ( Z p ) r freely acting on a finite CW-complexhomotopic to ( S n ) k suggested by P. E. Conner ([10]), the problem has made anessential progress (see [1], [7]–[8] and [18]). In the general case, the inequality (1.1)was conjectured by S. Halperin in [13] for the torus case, and by G. Carlsson in [9] † T. E. Panov informs of us that using a different method, Yury Ustinovsky has also recentlyproved the Halperin’s toral rank conjecture for the moment-angle complexes with the restrictionof natural tori actions, see arXiv:0909.1053. ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 3 for the p -torus case. So far, the conjecture holds if r ≤ r ≤ p -torus case (see [16]). Also, many authors have givencontributions to the conjecture in many different aspects. For more details, see,e.g., [2]–[3], [6] and [15].The paper is organized as follows. In Section 2 we study the basic structure of thealgebra 2 [ m ] ∗ and the basic properties of the Z / Z -valued M¨obius transform, andreview the notions of Stanley–Reisner face rings and their Tor-algebras. Section 3is the main part of this paper. We give the proof of the algebra–combinatoricsformula and estimate the lower bound of | supp( M ( f )) | therein. In Section 4 weintroduce the definitions of Z K and R Z K , and review the theorem of V. M. Buch-staber and T. E. Panov on the cohomology of Z K . In particular, we also calculatethe cohomology (as a graded k -module) of the generalized moment-angle complex Z ( D , S ) K , see Subsection 4.2 for the definition of Z ( D , S ) K . Finally we finish the proof ofTheorem 1.3 in Section 5.2. M¨obius transform and Stanley–Reisner face ring
An algebra over Z / Z . Let 2 [ m ] denote the power set of [ m ], which is theset of all subsets (including the empty set) of [ m ]. Then 2 [ m ] forms a poset withrespect to the inclusion ⊆ , and it is also a boolean algebra under the set operationsof union, intersection and complement relative to [ m ]. Let2 [ m ] ∗ = (cid:8) f (cid:12)(cid:12) f : 2 [ m ] −→ Z / Z = { , } (cid:9) . Then 2 [ m ] ∗ forms an algebra over Z / Z , where the addition is defined by ( f + g )( a ) = f ( a ) + g ( a ) and the multiplication is defined by ( f · g )( a ) = f ( a ) g ( a ) for a ∈ [ m ] .Given a function f ∈ [ m ] ∗ , definesupp( f ) := f − (1)which is called the support of f . Definition 2.1.
For each a ∈ [ m ] , the function δ a ∈ [ m ] ∗ defined by δ a ( b ) = ( b = a the a -function . For each i ∈ [ m ], the function x i ∈ [ m ] ∗ defined by x i ( a ) = 1 ⇔ i ∈ a for ∀ a ∈ [ m ] is called the i -th coordinate function . Lemma 2.1. { δ a | a ∈ [ m ] } forms a basis for [ m ] ∗ .Proof. This is because any f ∈ [ m ] ∗ can be expressed as f = X a ∈ [ m ] f ( a ) δ a = X a ∈ supp( f ) δ a . (cid:3) By 1 one denotes the constant function such that 1( a ) = 1 for all a in 2 [ m ] .Obviously, 1 = P a ∈ [ m ] δ a . For each a ∈ [ m ] , set µ a := (Q i ∈ a x i if a is nonempty1 if a is empty . XIANGYU CAO AND ZHI L¨U
Then it is easy to see that
Lemma 2.2.
Let a, b ∈ [ m ] . Then µ a ( b ) = 1 ⇔ a ⊆ b . Definition 2.2. f ∈ [ m ] ∗ is said to be nice if supp( f ) is an abstract simplicialcomplex on vertex set S a ∈ supp( f ) a ⊆ [ m ]. Note that an abstract simplicial complex K on a subset of [ m ] is a collection of subsets in [ m ] with the property that foreach a ∈ K , all subsets (including the empty set) of a belong to K . Each a ∈ K is called a simplex and has dimension | a | −
1. The dimension of K is defined asmax a ∈ K { dim a } .It is easy to see that f is nice if and only if for each a ∈ supp( f ), any subset b ⊆ a has the property f ( b ) = 1.Let F [ m ] = { f ∈ [ m ] ∗ | f is nice } , and K [ m ] the set of all abstract simplicialcomplexes on vertex set A where A runs over all possible subsets in [ m ]. Proposition 2.1.
All functions of F [ m ] bijectively correspond to all abstract sim-plicial complexes of K [ m ] .Proof. Clearly, f supp( f ) gives a bijection F [ m ] −→ K [ m ] , whose inverse is K P a ∈ K δ a . (cid:3) M¨obius transform.
Based upon Proposition 2.1, we shall carry out our workfrom the viewpoint of functional analysis.
Definition 2.3.
The map M : 2 [ m ] ∗ −→ [ m ] ∗ given by the formula M ( f )( a ) = X b ⊆ a f ( b )for all f ∈ [ m ] ∗ and a ∈ [ m ] is called the Z / Z -valued M¨obius transform . Lemma 2.3. M is a linear transform such that M = id . In particular, (2.1) M ( δ a ) = µ a for any a ∈ [ m ] . Consequently, M ( µ a ) = δ a .Proof. The linearity of M is obvious. To check that M = id, take f ∈ [ m ] ∗ , onehas that for any a ∈ [ m ] (2.2) M ( f )( a ) = X b ⊆ a X c ⊆ b f ( c ) = X c ⊆ a X b ∈ [ c,a ] f ( c ) = f ( a ) + X c ( a X b ∈ [ c,a ] f ( c )For every term in the latter sum of (2.2), from c ( a we see that [ c, a ] is a booleansubalgebra of 2 [ m ] which has 2 k elements for some k >
0. So the sum P b ∈ [ c,a ] f ( c ) =0 in Z / Z . Therefore M ( f )( a ) = f ( a ) for any a ∈ [ m ] , so M ( f ) = f as desired.The equation (2.1) is a direct calculation by Lemma 2.2. (cid:3) As a consequence of Lemmas 2.1 and 2.3, one has
Corollary 2.1. { µ a | a ∈ [ m ] } is also a basis of [ m ] ∗ .Remark . By definition of M , if f ( ∅ ) = 1 then M ( f )( ∅ ) = 1.In the next two subsections we shall review the Stanley–Reisner face rings andTor-algebras. Our main reference is the book by E. Miller and B. Sturmfels ([14]). ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 5 Stanley–Reisner face ring.
Now let f ∈ F [ m ] be a nice function such that K f = supp( f ) ∈ K [ m ] is an abstract simplicial complex on [ m ] (so S a ∈ K f a = [ m ]).Following the notions of [14], let k [ v ] = k [ v , ..., v m ] be the polynomial al-gebra over k on m indeterminates v = v , ..., v m . Each monomial in k [ v ] hasthe form of v a = v a · · · v a m m for a vector a = ( a , ..., a m ) ∈ N m of nonnega-tive integers. Thus, k [ v ] is N m -graded, i.e., k [ v ] is a direct sum L a ∈ N m k [ v ] a with k [ v ] a · k [ v ] b = k [ v ] a + b where k [ v ] a = k { v a } is the vector space over k ,spanned by v a . Generally, a k [ v ]-module M is N m -graded if M = L b ∈ N m M b and v a · M b ⊆ M a + b . Given a vector a ∈ N m , by k [ v ]( − a ) one denotes the free k [ v ]-module generated in degree a . So k [ v ]( − a ) is isomorphic to the ideal h v a i as N m -graded modules. Furthermore, a free N m -graded module of rank r is isomorphicto the direct sum k [ v ]( − a ) L · · · L k [ v ]( − a r ) for some vectors a , ..., a r ∈ N m .A monomial v a in k [ v ] is said to be squarefree if every coordinate of a is 0 or 1,i.e., a ∈ { , } m called a squarefree vector . Clearly, all elements in 2 [ m ] bijectivelycorrespond to all vectors in { , } m by mapping ξ : a ∈ [ m ] a ∈ { , } m , where a has entry 1 in the i -th place when i ∈ a , and 0 in all other entries. With thisunderstanding, for a ∈ [ m ] , one may write v a = Q i ∈ a v i . Then the Stanley–Reisnerideal of K f is defined as I K f = h v τ | τ K f i . Furthermore, the quotient ring k ( K f ) = k [ v ] /I K f is called the Stanley–Reisner face ring . Example . If K f = 2 [ m ] then k ( K f ) = k [ v ], and if K f = 2 [ m ] \ { [ m ] } then k ( K f ) = k [ v ] / h v [ m ] i . k ( K f ) is a finitely generated graded k [ v ]-module. Hilbert’s syzygy theorem tellsus that there exists a free resolution of k ( K f ) of length at most m . One knowsfrom [14, Section 1.4] that k ( K f ) is N m -graded and it has an N m -graded minimalfree resolution as follows(2.3) 0 ←− k ( K f ) ←− F φ ←− F ←− · · · ←− F h − φ h ←− F h ←− φ i is N m -graded degree-preserving. Since each F i isa free N m -graded k [ v ]-module, one may write F i = L a ∈ N m k [ v ]( − a ) β k ( Kf ) i, a where β k ( K f ) i, a ∈ N (see also [14, Section 1.5]). By [14, Corollary 1.40], if a ∈ N m is notsquarefree, then β k ( K f ) i, a = 0 for all i . Thus, we only need to consider those β k ( K f ) i, a with a ∈ { , } m . Throughout the following we shall write β k ( K f ) i,a := β k ( K f ) i, a where a ∈ [ m ] with ξ ( a ) = a . Definition 2.4 (cf. [14, Definition 1.29]) . The number β k ( K f ) i,a is called the ( i, a ) -thBetti number of k ( K f ).2.4. Tor-algebra of k ( K f ) . Applying the functor ⊗ k [ v ] k to the sequence (2.3),one may obtain the following chain complex of N m -graded k [ v ]-modules:0 ←− F ⊗ k [ v ] k φ ′ ←− F ⊗ k [ v ] k ←− · · · φ ′ h ←− F h ⊗ k [ v ] k ←− . Since the free resolution (2.3) is minimal, the differentials φ ′ i ’s become zero homo-morphisms. Then the i -th homology module of the above chain complex is ker φ ′ i Im φ ′ i +1 = XIANGYU CAO AND ZHI L¨U F i ⊗ k [ v ] k , denoted by Tor k [ v ] i ( k ( K f ) , k ). Namely, Tor k [ v ] i ( k ( K f ) , k ) = F i ⊗ k [ v ] k so dim k Tor k [ v ] i ( k ( K f ) , k ) = rank F i = X a ∈ [ m ] β k ( K f ) i,a . This also implies that for a ∈ N m with a
6∈ { , } m , Tor k [ v ] i ( k ( K f ) , k ) a = 0, and soTor k [ v ] i ( k ( K f ) , k ) can be decomposed into a direct sum M a ∈ [ m ] Tor k [ v ] i ( k ( K f ) , k ) a with dim k Tor k [ v ] i ( k ( K f ) , k ) a = β k ( K f ) i,a (see also [14, Lemma 1.32]). Furthermore,one has thatTor k [ v ] ( k ( K f ) , k ) = h M i =0 Tor k [ v ] i ( k ( K f ) , k ) = M i ∈ [0 ,h ] ∩ N ,a ∈ [ m ] Tor k [ v ] i ( k ( K f ) , k ) a which is a bigraded k [ v ]-module. Combining with the above arguments, this gives Proposition 2.2. h P i =0 dim k Tor k [ v ] i ( k ( K f ) , k ) = h P i =0 P a ∈ [ m ] β k ( K f ) i,a . M¨obius transform of abstract simplicial complexes and Bettinumbers of face rings
An algebra–combinatorics formula.
Following Subsections 2.3–2.4, nowlet us investigate the essential relationship between the M¨obius transform M ( f ) of f and the Betti numbers of the face ring k ( K f ) of K f . Theorem 3.1 (Algebra–combinatorics formula) . M ( f ) = h X i =0 X a ∈ [ m ] β k ( K f ) i,a δ a . Proof.
For any b ∈ [ m ] , the exact sequence (2.3) in degree b reads into0 ←− k D b ←− k d b, ←− k d b, ←− k d b, ←− · · · ←− k d b,h ←− D b = dim k k ( K f ) b and d b,i = dim k ( F i ) b . Since the above sequence isalso exact, we have that D b = P hi =0 ( − i d i,b . An easy observation shows that f ( b ) = dim k k ( K f ) b = D b , and d b,i = P a ⊆ b β k ( K f ) i,a (this is induced from F i = L a ∈ N m k [ v ]( − a ) β k ( Kf ) i, a ).Now let us work in integers modulo 2. We then have that D b = P hi =0 d i,b , andfurther f ( b ) = h X i =0 X a ⊆ b β k ( K f ) i,a = h X i =0 X a ∈ [ m ] β k ( K f ) i,a µ a ( b ) . So f = h X i =0 X a ∈ [ m ] β k ( K f ) i,a µ a . Applying M to the above equality and noting that M ( µ a ) = δ a , we arrive at therequired formula. (cid:3) ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 7 Corollary 3.2.
Let f ∈ [ m ] ∗ be a nice function such that K f = supp( f ) ∈ K [ m ] is an abstract simplicial complex on [ m ] . Then | supp( M ( f )) | ≤ h X i =0 X a ∈ [ m ] β k ( K f ) i,a . Proof.
From the formula of Theorem 3.1, one has that M ( f ) = h X i =0 X a ∈ [ m ] β k ( K f ) i,a δ a = X a ∈ [ m ] (cid:16) h X i =0 β k ( K f ) i,a (cid:17) δ a so for any a ∈ supp( M ( f )), P hi =0 β k ( K f ) i,a must be odd and nonnegative, and then P hi =0 β k ( K f ) i,a ≥
1. Therefore h X i =0 X a ∈ [ m ] β k ( K f ) i,a ≥ X a ∈ supp( M ( f )) h X i =0 β k ( K f ) i,a ≥ X a ∈ supp( M ( f )) | supp( M ( f )) | as desired. (cid:3) The estimation of the lower bound of | supp ( M ( f )) | . We shall upbuilda method of compressing supp( f ) to get the desired lower bound of | supp( M ( f )) | . Definition 3.1.
Fix k ∈ [ m ]. f ∈ F [ m ] is k -th extendable if(k-1) f ( { k } ) = 1;(k-2) M ( f ) · x k = 0 in 2 [ m ] ∗ .The linear transformation E k : 2 [ m ] ∗ −→ [ m ] ∗ determined by µ a µ a \{ k } is calledthe k -th compression-operator . f ∈ F [ m ] is said to be extendable if there is some k ∈ [ m ] such that f is k -th extendable; otherwise, f is said to be non-extendable .Introducing the map ǫ k : 2 [ m ] −→ [ m ] defined by a a ∪ { k } , we derive thefollowing formula for E k . Lemma 3.1.
For any f ∈ [ m ] ∗ we have E k ( f ) = f ◦ ǫ k . Proof.
It suffices to check that the formula E k ( µ a ) = µ a ◦ ǫ k holds for each a ∈ [ m ] .Indeed, take b ∈ [ m ] , we have that E k ( µ a )( b ) = 1 ⇔ a \ { k } ⊆ b ⇔ a ⊆ b ∪ { k } ⇔ µ a ( ǫ k ( b )) = 1 . Therefore, E k ( µ a ) = µ a ◦ ǫ k as desired. (cid:3) Proposition 3.1.
Fix k ∈ [ m ] . If f ∈ F [ m ] satisfies f ( { k } ) = 1 , then E k ( f ) ∈ F [ m ] and supp( E k ( f )) ⊆ supp( f ) .Proof. For any pair a ⊆ b in 2 [ m ] , we have that ǫ k ( a ) ⊆ ǫ k ( b ). So if E k ( f )( b ) = f ( ǫ k ( b )) = 1, then f ( ǫ k ( a )) = 1 since f ∈ F [ m ] , and so E k ( f )( a ) = 1. Also, f ( { k } ) = 1 implies that E k ( f )( ∅ ) = f ( ∅ ∪ { k } ) = 1. Thus, E k ( f ) is nice.For any a ∈ [ m ] , if E k ( f )( a ) = 1 then by Lemma 3.1 f ( ǫ k ( a )) = 1, so f ( a ) = 1since a ⊆ ǫ k ( a ) and f ∈ F [ m ] . Hence, supp( E k ( f )) ⊆ supp( f ) as desired. (cid:3) XIANGYU CAO AND ZHI L¨U
Now let us look at the composition transformation
M ◦ E k ◦ M =: ˆ E k . For any a ∈ [ m ] , one has(3.1) ˆ E k ( δ a ) = M ◦ E k ◦ M ( δ a ) = M ◦ E k ( µ a ) = M ( µ a \{ k } ) = δ a \{ k } . Note also that since M = id, M ◦ E k = ˆ E k ◦ M . Lemma 3.2.
For any g ∈ [ m ] ∗ and k ∈ [ m ] , ˆ E k ( g ) x k = 0 .Proof. Write g = P a ∈ supp( g ) δ a . Since ˆ E k is linear and ˆ E k ( δ a ) = δ a \{ k } for any a ∈ [ m ] , it follows that ˆ E k ( g ) = P a ∈ supp( g ) δ a \{ k } . Obviously, for any a ∈ [ m ] , δ a \{ k } x k = 0. Thus, ˆ E k ( g ) x k = 0 as desired. (cid:3) Corollary 3.3.
Let k ∈ [ m ] . If f ∈ [ m ] ∗ satisfies M ( f ) x k = 0 , then f = E k ( f ) .Proof. Suppose that f = E k ( f ). Applying M to both sides, we get M ( f ) = M ( E k ( f )) = ˆ E k ( M ( f )). Write g = M ( f ). Then g = ˆ E k ( g ). Multiplied by x k onthe two sides of g = ˆ E k ( g ), we have that gx k = ˆ E k ( g ) x k . Since gx k = M ( f ) x k = 0,we have ˆ E k ( g ) x k = 0, a contradiction by Lemma 3.2. (cid:3) Proposition 3.2.
Let f ∈ [ m ] ∗ . Then for each k ∈ [ m ] , | supp( ˆ E k ( f )) | ≤ | supp( f ) | . Proof.
Let A = (cid:8) a ∈ [ m ] (cid:12)(cid:12) k / ∈ a, a ∈ supp( f ) (cid:9) and B = (cid:8) a ∈ [ m ] (cid:12)(cid:12) k / ∈ a, ǫ k ( a ) ∈ supp( f ) (cid:9) . Then we have f = X a ∈ supp( f ) δ a = X a ∈ supp( f ) k/ ∈ a δ a + X a ∈ supp( f ) k ∈ a δ a = X a ∈ A δ a + X a ∈ B δ ǫ k ( a ) and by (3.1)ˆ E k ( f ) = X a ∈ A δ a \{ k } + X a ∈ B δ ǫ k ( a ) \{ k } = X a ∈ A δ a + X a ∈ B δ a = X a ∈ A △ B δ a where A △ B = ( A \ B ) ∪ ( B \ A ). Now | supp( ˆ E k ( f )) | = | A △ B | ≤ | A | + | B | = | supp( f ) | as desired. (cid:3) Remark . Observe that for any f ∈ F [ m ] , whenever f is k -th extendable forsome k ∈ [ m ], by Proposition 3.1 and Corollary 3.3 we obtain that E k ( f ) ∈ F [ m ] and supp( E k ( f )) ( supp( f ). In addition, since ( M ◦ E k )( f ) = ( ˆ E k ◦ M )( f ), byProposition 3.2 one has that | supp( M ( E k ( f ))) | ≤ | supp( M ( f )) | . We replace f with E k ( f ) and repeat the above process whenever possible, so as to get a sequenceof functions in F [ m ] with strictly decreasing support. This process must end aftera finite number of steps, giving finally a f ∈ F [ m ] that is non-extendable withsupp( f ) ⊆ supp( f ) and | supp( M ( f )) | ≤ | supp( M ( f )) | . It remains to characterizesuch a non-extendable f ∈ F [ m ] . Proposition 3.3.
Let f ∈ F [ m ] . Then f is non-extendable if and only if there issome a ∈ [ m ] such that supp( f ) = 2 a ( i.e., f = P b ⊆ a δ b ) . ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 9 Proof.
Suppose that f is non-extendable. Let a = (cid:8) k ∈ [ m ] (cid:12)(cid:12) f ( { k } ) = 1 (cid:9) . If a = ∅ , obviously we have f = δ ∅ . Assume that a is non-empty. Given an element b ∈ [ m ] , if f ( b ) = 1, since f ∈ F [ m ] , then for any k ∈ b, f ( { k } ) = 1 so k ∈ a and b ⊆ a . Since f is non-extendable, M ( f ) x k = 0 for any k ∈ a . Then we seefrom M ( f ) = P b ∈ supp( M ( f )) δ b that for any b ∈ supp( M ( f )), b ∩ a = ∅ . Since M ( f )( ∅ ) = 1, we have that ∅ ∈ supp( M ( f )). Furthermore f ( a ) = M ( f )( a ) = M (cid:16) X b ∈ supp( M ( f )) δ b (cid:17) ( a ) = X b ∈ supp( M ( f )) µ b ( a ) = µ ∅ ( a ) = 1 . Since f ∈ F [ m ] , it follows that for any subset b ⊆ a , f ( b ) = 1. Therefore, for b ∈ [ m ] f ( b ) = 1 ⇔ b ⊆ a . This implies that supp( f ) = 2 a = { b ∈ [ m ] (cid:12)(cid:12) b ⊆ a } .Conversely, suppose that f = P b ⊆ a δ b for some a ∈ [ m ]. If a = [ m ], then f = 1 = µ ∅ so M ( f ) = δ ∅ . Moreover, for any k ∈ a , M ( f ) x k = 0 so f is non-extendable. If a = ∅ , obviously f is non-extendable. Assume that a = [ m ] , ∅ .Then an easy argument shows that f = Y i ∈ [ m ] \ a (1 + x i ) = X b ⊆ [ m ] \ a µ b . Applying M to the above equality, it follows that M ( f ) = P b ⊆ [ m ] \ a δ b . Now forany k ∈ a and any b ⊆ [ m ] \ a , we have δ b x k = 0 so M ( f ) x k = 0. This meansthat f is also non-extendable. (cid:3) From the proof of Proposition 3.3, we easily see that
Corollary 3.4.
Let a ∈ [ m ] . Then f = P b ⊆ a δ b if and only if M ( f ) = P b ⊆ [ m ] \ a δ b ( i.e., supp( f ) = 2 a if and only if supp( M ( f )) = 2 [ m ] \ a ) . In this case, | supp( M ( f )) | =2 m −| a | . We now summarize the above arguments as follows.
Theorem 3.5.
For any f ∈ F [ m ] , there exists some a ∈ supp( f ) such that | supp( M ( f )) | ≥ m −| a | . Remark . The interested readers are invited to see a simple fact that f ∈ F [ m ] canbe compressed by compression-operators into a non-extendable f with supp( f ) =2 a if and only if a is a maximal element in supp( f ) as a poset. This result willnot be used later in this article.4. Moment-angle complexes and their cohomologies
Let K be an abstract simplicial complex on vertex set [ m ]. Let ( X, W ) be apair of topological spaces with W ⊂ X . Following [5, Construction 6.38], for eachsimplex σ in K , set B σ ( X, W ) = m Y i =1 A i such that A i = ( X if i ∈ σW if i ∈ [ m ] \ σ. Then one can define the following subspace of the product space X m : K ( X, W ) = [ σ ∈ K B σ ( X, W ) ⊂ X m . Moment-angle complexes.
When the pair (
X, W ) is chosen as ( D , S ), Z K := K ( D , S ) ⊂ ( D ) m is called the moment-angle complex on K where D = (cid:8) z ∈ C (cid:12)(cid:12) | z | ≤ (cid:9) is the unitdisk in C , and S = ∂D . Since ( D ) m ⊂ C m is invariant under the standard actionof T m on C m given by (cid:0) ( g , ..., g m ) , ( z , ..., z m ) (cid:1) ( g z , ..., g m z m ) , ( D ) m admits a natural T m -action whose orbit space is the unit cube I m ⊂ R m ≥ .The action T m y ( D ) m then induces a canonical T m -action Φ on Z K .When the pair ( X, W ) is chosen as ( D , S ), R Z K := K ( D , S ) ⊂ ( D ) m is called the real moment-angle complex on K where D = (cid:8) x ∈ R (cid:12)(cid:12) | x | ≤ (cid:9) = [ − , R , and S = ∂D = {± } . Similarly, ( D ) m ⊂ R m is invariantunder the standard action of ( Z ) m on R m given by (cid:0) ( g , ..., g m ) , ( x , ..., x m ) (cid:1) ( g x , ..., g m x m ) . Thus ( D ) m admits a natural ( Z ) m -action whose orbit space is also the unit cube I m ⊂ R m ≥ , where Z = {− , } is the group with respect to multiplication. Fur-thermore, the action ( Z ) m y ( D ) m also induces a canonical ( Z ) m -action Φ R on R Z K .Let P K be the cone on the barycentric subdivision of K . Since the cone on thebarycentric subdivision of a k -simplex is combinatorially equivalent to the standardsubdivision of a ( k +1)-cube, P K is naturally a cubical complex and it is decomposedinto cubes indexed by the simplices of K . Then one knows from [5] and [11] thatboth T m -action Φ on Z K and ( Z ) m -action Φ R on R Z K have the same orbit space P K . Example . When K = 2 [ m ] , Z K = ( D ) m and R Z K = ( D ) m . When K =2 [ m ] \ { [ m ] } , Z K = S m − and R Z K = S m − . Remark . In general, Z K and R Z K are not manifolds. However, if K is a simplicialsphere, then both Z K and R Z K are closed manifolds (see [5, Lemma 6.13]).4.2. Cohomology.
V. M. Buchstaber and T. E. Panov in [5, Theorem 7.6] havecalculated the cohomology of Z K (see also [15, Theorem 4.7]). Their result is statedas follows. Theorem 4.1 (Buchstaber–Panov) . As k -algebras, H ∗ ( Z K ; k ) ∼ = Tor k [ v ] ( k ( K ) , k ) where k ( K ) = k [ v ] /I K = k [ v , ..., v m ] /I K with deg v i = 2 . ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 11 Here we shall calculate the cohomologies of a class of generalized moment-anglecomplexes. For this, we begin with the notion of the generalized moment-anglecomplex, due to N. Strickland, cf. [4] and [12]. Given an abstract simplicial complex K on [ m ], let ( X, W ) = { ( X i , W i ) } mi =1 be m pairs of CW-complexes with W i ⊂ X i .Then the generalized moment-angle complex is defined as follows: K ( X, W ) = [ σ ∈ K B σ ( X, W ) ⊂ m Y i =1 X i where B σ ( X, W ) = Q mi =1 H i and H i = ( X i if i ∈ σW i if i ∈ [ m ] \ σ. Now take (
X, W ) = ( D , S ) = { ( D i , S i ) } mi =1 with each CW-complex pair ( D i , S i )subject to the following conditions:(1) D i is acyclic, that is, e H j ( D i ) = 0 for any j .(2) There exists a unique κ i such that e H κ i ( S i ) = Z and e H j ( S i ) = 0 for any j = κ i .Then our objective is to calculate the cohomology of Z ( D , S ) K := K ( D , S ) = [ σ ∈ K B σ ( D , S ) ⊂ m Y i =1 D i . First, for each i ∈ [ m ], it follows immediately from the long exact sequence of( D i , S i ) that0 = e H κ i ( D i ; k ) −→ e H κ i ( S i ; k ) ∼ = −→ e H κ i +1 (( D i , S i ; k ) −→ e H κ i +1 ( D i ; k ) = 0 . On the cellular cochain level, one has the following short exact sequence0 −→ D ∗ ( D i , S i ; k ) j ∗ −→ D ∗ ( D i ; k ) i ∗ −→ D ∗ ( S i ; k ) −→ D k ( D i , S i ; k ) can be considered as a subgroup of D k ( D i ; k ), so j ∗ is aninclusion. By the zig-zag lemma, one can choose a κ i -cochain x i of D κ i ( D i ; k ) suchthat • i ∗ ( x i ) represents a generator of e H κ i ( S i ; k ). • dx i ∈ ker i ∗ so j ∗ ( dx i ) = dx i ∈ D κ i +1 ( D i , S i ; k ) ⊆ D κ i +1 ( D i ; k ) generates e H κ i +1 (( D i , S i ; k ), where d is the coboundary operator of D ∗ ( D i ; k ).Write x (1) i = x i and x (2) i = dx i , and let x (0) i denote the constant 0-cochain 1 in D ( D i ; k ). Obviously, x (0) i , x (1) i and x (2) i are linearly independent in D ∗ ( D i ; k ) asa k -vector space.Now let us work in the cellular cochain complex D ∗ ( Q mi =1 D i ; k ) of the productspace Q mi =1 D i . Let Ω ∗ be the vector subspace of D ∗ ( Q mi =1 D i ; k ) spanned by thefollowing cross products x ( k )1 × · · · × x ( k m ) m , k i ∈ { , , } . An easy observation shows that Ω ∗ is a cochain subcomplex of D ∗ ( Q mi =1 D i ; k ), and (cid:8) x ( k )1 × · · · × x ( k m ) m (cid:12)(cid:12) k i ∈ { , , } (cid:9) forms a basis of Ω ∗ as a vector space over k since x (0) i , x (1) i and x (2) i are linearly independent in D ∗ ( D i ; k ). For a convenience, we write each basis element x ( k )1 × · · · × x ( k m ) m of Ω ∗ as the following form x ( τ,σ ) where x = x ( k )1 , ..., x ( k m ) m , τ = { i (cid:12)(cid:12) k i = 1 } and σ = { i (cid:12)(cid:12) k i = 2 } . In particular, if τ = σ = ∅ , then x ( ∅ , ∅ ) = x (0)1 × · · · × x (0) m . Thus, Ω ∗ can be expressed asΩ ∗ = Span (cid:8) x ( τ,σ ) (cid:12)(cid:12) τ, σ ⊆ [ m ] with τ ∩ σ = ∅ (cid:9) . Next by Φ K we denote the compositionΩ ∗ ֒ → D ∗ ( m Y i =1 D i ; k ) l ∗ −→ D ∗ ( Z ( D , S ) K ; k )where the latter map l ∗ is induced by the inclusion l : Z ( D , S ) K ֒ → Q mi =1 D i , and it issurjective. Set S K = Span (cid:8) x ( τ,σ ) ∈ Ω ∗ (cid:12)(cid:12) σ K (cid:9) . Clearly it is a cochain subcomplex of Ω ∗ . Lemma 4.1. S K ⊆ ker Φ K . Furthermore, Φ K can induce a cochain map Ω ∗ /S K −→ D ∗ ( Z ( D , S ) K ; k ) , also denoted by Φ K .Proof. Let x ( τ,σ ) be a basis element in S K ⊂ D ∗ ( Q mi =1 D i ; k ). For any productcell e = e × · · · × e m ⊂ Z ( D , S ) K ⊆ Q mi =1 D i , there must be some σ ′ ∈ K suchthat e ⊂ B σ ′ ( D , S ), where each e i can represent a generator in the celluar chaingroup D dim e i ( D i ; k ). In addition, it is easy to see that e can also be regarded asa generator of the cellular chain complex D ∗ ( Z ( D , S ) K ; k ) l ∗ ֒ → D ∗ ( Q mi =1 D i ; k ) where l ∗ is the inclusion induced by l : Z ( D , S ) K ֒ → Q mi =1 D i . Since σ K , σ is non-empty. Moreover, there is some i ∈ σ \ σ ′ such that e i ⊂ S i ⊂ D i and thefactor x (2) i ∈ D κ i +1 ( D i , S i ; k ) ⊂ D κ i +1 ( D i ; k ) in x ( τ,σ ) , together yielding that h x (2) i , e i i = 0. Therefore, h x ( τ,σ ) , l ∗ ( e ) i = h x ( τ,σ ) , e i = 0 by the definition of crossproduct. Furthermore, we have that the value of Φ K ( x ( τ,σ ) ) on e is h Φ K ( x ( τ,σ ) ) , e i = h x ( τ,σ ) ◦ l ∗ , e i = h x ( τ,σ ) , l ∗ ( e ) i = 0so Φ K ( x ( τ,σ ) ) = 0 in D ∗ ( Z ( D , S ) K ; k ), as desired. (cid:3) By Ω ∗ ( K ) we denote the quotient Ω ∗ /S K . Let L be a subcomplex of K . Thenwe obtain a pair ( Z ( D , S ) K , Z ( D , S ) L ) of CW-complexes. Now since S K ⊆ S L , we have ashort exact sequence(4.1) 0 −→ ker( π ∗ ) −→ Ω ∗ ( K ) π ∗ −→ Ω ∗ ( L ) −→ π ∗ is induced by the natural inclusion π : S K ֒ → S L . By Ω ∗ ( K, L ) we de-note the kernel ker π ∗ . It is easy to see that two cochain maps Φ K : Ω ∗ ( K ) −→ D ∗ ( Z ( D , S ) K ; k ) and Φ L : Ω ∗ ( L ) −→ D ∗ ( Z ( D , S ) L ; k ) give a cochain map Φ ( K,L ) :Ω ∗ ( K, L ) −→ D ∗ ( Z ( D , S ) K , Z ( D , S ) L ; k ) such that the following diagram commutes0 / / Ω ∗ ( K, L ) Φ ( K,L ) (cid:15) (cid:15) / / Ω ∗ ( K ) Φ K (cid:15) (cid:15) π ∗ / / Ω ∗ ( L ) Φ L (cid:15) (cid:15) / / / / D ∗ ( Z ( D , S ) K , Z ( D , S ) L ; k ) / / D ∗ ( Z ( D , S ) K ; k ) / / D ∗ ( Z ( D , S ) L ; k ) / / . ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 13 Furthermore, we may obtain a homomorphism between two long exact cohomologysequences given by two short exact sequences above.
Proposition 4.1.
For any K ∈ K [ m ] , Φ K induces an isomorphism H ∗ (Ω ∗ ( K ); k ) ∼ = −→ H ∗ ( Z ( D , S ) K ; k ) as graded k -modules.Proof. First observe that for K = {∅} , Ω ∗ ( K ) is spanned by { x ( τ, ∅ ) (cid:12)(cid:12) τ ⊆ [ m ] } withzero coboundary operator. On the other hand, if K = {∅} then Z ( D , S ) K = Q mi =1 S i .By the K¨unneth formula, the above set is not but a basis of H ∗ ( Z ( D , S ) K ; k ) as agraded k -module (if we view the elements of the set as cohomological classes).Thus, clearly Φ K induces an isomorphism in this case.Next we proceed inductively by considering a pair of abstract simplicial com-plexes ( K, L ) where K = L ⊔ { σ } for some simplex σ (which is a maximal elementof K as a poset). Hence ( Z ( D , S ) K , Z ( D , S ) L ) is a pair of CW-complexes, which has byexcision the same cohomology as ( Z ( D , S )2 σ , Z ( D , S )2 σ \ σ ). This pair ( Z ( D , S )2 σ , Z ( D , S )2 σ \ σ ) is inturn homeomorphic to Y i ∈ [ m ] \ σ S i × (cid:16) Y i ∈ σ D i , A ( Y i ∈ σ D i ) (cid:17) where A ( Q i ∈ σ D i ) = ( S i × D i × · · · × D i s ) ∪ · · · ∪ ( D i × · · · × D i s − × S i s ) with σ = { i , ..., i s (cid:12)(cid:12) i < · · · < i s } . By relative K¨unneth formula, its cohomology with k coefficients is isomorphic toSpan { x ( τ,σ ) (cid:12)(cid:12) τ ⊆ [ m ] with τ ∩ σ = ∅} as graded k -modules. On the other hand, we see easily from the short exact se-quence (4.1) that Ω ∗ ( K, L ) = ker π ∗ is exactly equal to the cochain complexSpan { x ( τ,σ ) (cid:12)(cid:12) τ ⊆ [ m ] with τ ∩ σ = ∅} with zero coboundary operator. It then follows that Φ ( K,L ) induces an isomor-phism H ∗ (Ω ∗ ( K, L ); k ) ∼ = −→ H ∗ ( Z ( D , S ) K , Z ( D , S ) L ; k ) as graded k -modules. Induc-tively, now we may assume that Φ L induces an isomorphism H ∗ (Ω ∗ ( L ); k ) −→ H ∗ ( Z ( D , S ) L ; k ) as graded k -modules. Hence we may conclude that the same holdsfor H ∗ (Ω ∗ ( K ); k ) −→ H ∗ ( Z ( D , S ) K ; k ) by the five-lemma. This completes the induc-tion and the proof of Proposition 4.1. (cid:3) Now let us return back to study the complex (Ω ∗ ( K ) , d ). First, we may imposea { , } m -graded (or 2 [ m ] -graded) structure on Ω ∗ ( K ), by defining for a ∈ [ m ] Ω ∗ ( K ) a := Span (cid:8) x ( τ,σ ) (cid:12)(cid:12) τ ⊆ [ m ] , σ ∈ K with τ ∪ σ = a, τ ∩ σ = ∅ (cid:9) . Then, clearly Ω ∗ ( K ) = L a ∈ [ m ] Ω ∗ ( K ) a . Furthermore, given a basis element x ( τ,σ ) ∈ Ω ∗ ( K ) a with τ = a \ σ , by a direct calculation we have that d ( x ( a \ σ,σ ) ) = X k ∈ a \ σσ ∪{ k }∈ K ǫ k x ( a \ ( σ ∪{ k } ) ,σ ∪{ k } )4 XIANGYU CAO AND ZHI L¨U which still belongs to Ω ∗ ( K ) a , where ǫ k = ±
1. So (Ω ∗ ( K ) a , d ) has also a cochaincomplex structure. This means that Ω ∗ ( K ) is a bigraded k -module. Also, clearlythe basis of Ω ∗ ( K ) a is indexed by K | a where K | a = { σ ∈ K (cid:12)(cid:12) σ ⊆ a } . Lemma 4.2.
For each a ∈ [ m ] , (Ω ∗ ( K ) a , d ) is isomorphic to the coaugmentedcochain complex ( C ∗ ( K | a ; k ) , d ′ ) as cochain complexes. Furthermore, H ∗ (Ω ∗ ( K ) a ; k ) ∼ = e H ∗ ( K | a ; k ) as graded k -modules. Lemma 4.2 is a (dualized) consequence of the following general result.
Lemma 4.3.
Let K be an abstract simplicial complex on a finite set. Let V ( K ) be avector space over k with a K -indexed basis { v σ | σ ∈ K } , and let ι : V ( K ) −→ V ( K ) be a linear map such that ι = 0 and ι ( v σ ) = P k ∈ σ ε k v σ \{ k } where ε k = ± .Then there is an isomorphism f : V ( K ) −→ C ∗ ( K ; k ) as k -vector spaces with form f : v σ ε σ σ such that f ◦ ι = ∂ ◦ f , where ε σ = ± and C ∗ ( K ; k ) is the ordinarychain complex over k of K with the boundary operator ∂ .Proof. We proceed inductively. For K = {∅} , V ( K ) = Span { v ∅ } ∼ = k with ι = 0and C ∗ ( K ; k ) = Span {∅} ∼ = k with ∂ = 0, so clearly we have such a f . Now for anarbitrary K = {∅} , take a maximal element σ of K (as a poset) so that L = K \{ σ } is a subcomplex of K . The subspace V ( K ) | L = Span { v σ | σ ∈ L } is invariant under ι . So we can apply induction hypothesis to ( V ( K ) | L , ι ), yielding an isomorphism f : V ( K ) | L −→ C ∗ ( L ; k ) by v σ ε σ σ such that f ◦ ι = ∂ ◦ f . Now observe that ι ( v σ ) = P k ∈ σ ε k v σ \{ k } ∈ V ( K ) | L , so f ( ι ( v σ )) = P k ∈ σ ε k ε σ \{ k } ( σ \ { k } )which is in the chain group C | σ |− (2 σ ; k ) ⊂ C | σ |− ( L ; k ), and ( ∂ ◦ f )( ι ( v σ )) =( f ◦ ι )( ι ( v σ )) = f ( ι ( v σ )) = 0, i.e., f ( ι ( v σ )) ∈ ker ∂ . Since C ∗ (2 σ ; k ) is acyclicand C | σ |− (2 σ ; k ) = Span { σ } , we have f ( ι ( v σ )) = ∂ ( nσ ) for some n ∈ k .However, ∂ ( nσ ) = n∂ ( σ ) so n∂ ( σ ) = P k ∈ σ ε k ε σ \{ k } ( σ \ { k } ). This forces n tobe ±
1. We can then extend f to f : V ( K ) −→ C ∗ ( K ; k ) by defining v σ nσ ,so that we have f ( ι ( v σ )) = f ( ι ( v σ )) = ∂ ( nσ ) = ∂ ( f ( v σ )) . Hence f ◦ ι = ∂ ◦ f in V ( K ). The induction step is finished, proving the lemma. (cid:3) The famous Hochster formula tells us (see [14, Corollary 5.12]) that for each a ∈ [ m ] , e H | a |− i − ( K | a ; k ) ∼ = Tor k [ v ] i ( k ( K ) , k ) a . We know by Lemma 4.2 that each class of e H | a |− i − ( K | a ; k ) may be understood asone of H ∗ (Ω ∗ ( K ) a ; k ), represented by a linear combination of the elements of theform x ( a \ σ,σ ) ∈ Ω ∗ ( K ) a with | σ | = | a | − i ; so by Proposition 4.1 it corresponds to acohomological class of degree | σ | + P k ∈ a κ k = − i + P k ∈ a ( κ k + 1) in H ∗ ( Z ( D , S ) K ; k ).To sum up, it follows that for each n ≥ H n ( Z ( D , S ) K ; k ) ∼ = M a ∈ m ] − i + P k ∈ a ( κk +1)= n Tor k [ v ] i ( k ( K ) , k ) a . Combining with all arguments above, we conclude that
Theorem 4.2.
As graded k -modules, H ∗ ( Z ( D , S ) K ; k ) ∼ = Tor k [ v ] ( k ( K ) , k ) . Together with Proposition 2.2 and Theorem 4.2, we obtain that ¨OBIUS TRANSFORM, MOMENT-ANGLE COMPLEXES AND H–C CONJECTURE 15
Corollary 4.3. P i dim k H i ( Z ( D , S ) K ; k ) = h P i =0 P a ∈ [ m ] β k ( K ) i,a . Remark . It should be pointed out that here we merely determine the k -modulestructure of H ∗ ( Z ( D , S ) K ; k ). Of course, this is enough for our purpose in this paper.Observe that if there are two i, j ∈ [ m ] with i = j such that κ i and κ j are even,then for x (2) i , x (2) j ∈ Ω ∗ ( K ), x (2) i × x (2) j = − x (2) j × x (2) i . This means that in this case,if k is not a field of characteristic 2, then H ∗ (Ω ∗ ( K ); k ) cannot be isomorphic toTor k [ v ] ( k ( K ) , k ) as k -algebras since k ( K ) is a commutative ring. Even when k is afield of characteristic 2, there is still some nuance preventing us from simply extend-ing the ring structure result (4.1) of Buchstaber and Panov to the case of, say R Z K ;Indeed, in this case x (1) i would be a 0-cochain, which satisfies x (1) i ∪ x (1) i = x (1) i ,whereas in the cases when κ i > x (1) i ∪ x (1) i would be instead zero element in H ∗ ( S i ; k ). Nevertheless, our calculation of the module structure actually representsany cohomological class in H ∗ ( Z ( D , S ) K ; k ) as a sum of x ( τ,σ ) ’s via the isomorphism H ∗ (Ω ∗ ( K ); k ) ∼ = H ∗ ( Z ( D , S ) K ; k ), from which we may also figure out the cohomo-logical equivalence relation amongst such sums; since the cup product of pairs ofthese elements is clear, in a certain sense we should have also determined the ringstructure of H ∗ ( Z ( D , S ) K ; k ). In other words, let k ( K ) = k [ v ] /I K = k [ v , ..., v m ] /I K be the Stanley–Reisner face ring of K with deg v i = κ i + 1. Then it should bereasonable to conjecture that the following results hold: • If all κ i ’s are odd, then H ∗ ( Z ( D , S ) K ; k ) ∼ = Tor k [ v ] ( k ( K ) , k ) as k -algebras. • If κ i > i ∈ [ m ], then H ∗ ( Z ( D , S ) K ; k ) ∼ = Tor k [ v ] ( k ( K ) , k ) as k -algebras. • In general, H ∗ ( Z ( D , S ) K ; k ) ∼ = H [ H ∗ ( Q mi =1 S i ; k ) ⊗ k [ v ] k ( K )] as k -algebras.5. Application to the free actions on Z K and R Z K First we prove a useful lemma.
Lemma 5.1.
Let K ∈ K [ m ] be an abstract simplicial complex on vertex set [ m ] ,and let H ( resp. H R ) be a rank r subtorus of T m ( resp. ( Z ) m ) . If H ( resp. H R ) can freely act on Z K ( resp. R Z K ) , then r ≤ m − dim K − .Proof. It is well-known that H (resp. H R ) can freely act on Z K (resp. R Z K ) ifand only if for any point z (resp. x ) of Z K (resp. R Z K ), H ∩ G z (resp. H R ∩ G x )is trivial, where G z (resp. G x ) is the isotropy subgroup at z (resp. x ). Supposethat r > m − dim K −
1. Take a ∈ K with | a | = dim K + 1. Without the loss ofgenerality, assume that a = { , ..., | a |} . Then we see that Z K (resp. R Z K ) containsthe point of the form z = (0 , ..., , z | a | +1 , ..., z m ) (resp. x = (0 , ..., , x | a | +1 , ..., x m )).It is easy to see that the isotropy subgroup G z (resp. G x ) has rank at least | a | ,so the intersection H ∩ G z (resp. H R ∩ G x ) cannot be trivial. This contradictionmeans that r must be equal to or less than m − dim K − (cid:3) Now let us use the preceding results to complete the proof of Theorem 1.3.
Proof Theorem 1.3 . Let f = P a ∈ K δ a ∈ F [ m ] such that supp( f ) = K . If f = 1(i.e., K = 2 [ m ] ), then Z K = ( D ) m (resp. R Z K = ( D ) m ). However, any properlynontrivial subtorus of T m (resp. ( Z ) m ) cannot freely act on ( D ) m (resp. ( D ) m )since the point (0 , ...,
0) is always a fixed point. Thus we may assume that f = 1. By Theorem 3.5, there exists some a ∈ [ m ] with a = [ m ] such that a ∈ supp( f ) = K and | supp( M ( f )) | ≥ m − n where n = | a | . Since a ∈ K , we have that n ≤ dim K +1.So by Lemma 5.1 it follows that n ≤ m − r and r ≤ m − n . Combining withTheorem 3.5 and Corollaries 3.2 and 4.3 together gives2 r ≤ m − n ≤ | supp( M ( f )) | ≤ X i dim k H i ( Z K ; k ) = X i dim k H i ( R Z K ; k )as desired. (cid:3) Acknowledgements.
The authors are grateful to M. Franz, M. Masuda, T. E. Panov,V. Puppe and L. Yu for their comments and suggestions.
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