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Buchstaber numbers and classical invariants of simplicial complexes

Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus actions on moment-angle complexes. In the paper we study the relation between Buchstaber invariants and classical invariants of simplicial complexes such as bigraded Betti numbers and chromatic invariants. The following two statements are proved. (1) There exists a simplicial complex U with different real and ordinary Buchstaber invariants. (2) There exist two simplicial complexes with equal bigraded Betti numbers and chromatic numbers, but different Buchstaber invariants. To prove the first theorem we define Buchstaber number as a generalized chromatic invariant. This approach allows to guess the required example. The task then reduces to a finite enumeration of possibilities which was done using GAP computational system. To prove the second statement we use properties of Taylor resolutions of face rings.


Introduction
Let K be a simplicial complex on a set of vertices [m] = {1, 2, . . . , m}. In toric topology a special topological space, called moment-angle complex, is associated to K. Definition 1.1 (Moment-angle complex [5,6]).   Constructions in toric topology, in particular moment-angle complexes, give rise to interesting and nontrivial invariants of simplicial complexes. Note that the actions T m Z K and Z m 2 R Z K are not free if K has at least one nonempty simplex. The main objects of this paper are Buchstaber invariants measuring the degree of symmetry of moment-angle complexes.
(1) The (ordinary) Buchstaber invariant s(K) of a simplicial complex K is the maximal rank of toric subgroups G ⊂ T m for which the restricted action G Z K is free. (2) The real Buchstaber invariant s R (K) is the maximal rank of subgroups G ⊂ Z m 2 for which the restricted action G R Z K is free.
Here "rank of subgroup G ⊂ Z m 2 " means the dimension of G as a vector subspace over the field of two elements. This finite field will also be denoted by Z 2 .
Several approaches to the study of Buchstaber invariants are developed up to date [21,22,12,13,16]. We refer to [14] for the comprehensive review of this field. In this paper we study the connection of Buchstaber invariants with each other and with other invariants of simplicial complexes.
Generally, there is a bound In toric topology the case s(K) = s R (K) = m − dim K − 1 is the most important; it appears quite often. Still there are many examples of K for which 1 < s R (K) < m − dim K − 1 or 1 < s(K) < m − dim K − 1. It is always very difficult to compute s(K) for such examples (Section 3 contains an example of such computation). The real invariant s R (K) is easier because its calculation allows computer-aided analysis. Thus an important question is: whether s(K) = s R (K) for any complex K? The answer is negative.
There exists a simplicial complex U of dimension 3 such that s(U ) = s R (U ).
Remark 1.3. Theorem 1 was announced without a proof in [2]. The proof was published later in [3], but unfortunately, that issue of the journal was not published in English. We provide the proof here (Section 3).
The second block of questions asks about the relation between Buchstaber invariants and other well-studied invariants. If A(·) is an invariant (possibly, a set of invariants) of a simplicial complex, then the general question is: There are several natural candidates for A(·): • Chromatic number γ(K) or its generalizations; • f -vector (or, equivalently, h-vector) of K; • Topological characteristics of K, e.g. Betti numbers; • Topological characteristics of the moment-angle complex Z K .
Classical chromatic number γ(K) on itself is too weak invariant for rigidity question 1.4 to make sense. On the other hand, Buchstaber invariants can themselves be considered as generalized chromatic invariants (see Section 2). N. Erokhovets [11,12] proved that Buchstaber invariants are not determined by the f -vector and the chromatic number. More precisely, he constructed two simplicial polytopes, whose boundaries have equal f -vectors and chromatic numbers, but Buchstaber invariants are different.
The cohomology ring of a moment-angle complex is the subject of intensive study during last fifteen years. It is known [5,15] that, the Tor-algebra of a Stanley-Reisner ring. The dimensions of graded components , k). are called bigraded Betti numbers of K. In general they depend on the ground field k. These invariants represent a lot of information about K [26,6]. In particular, from bigraded Betti numbers it is possible to extract: the h-vector of K; the ordinary Betti numbers of K and the ordinary Betti numbers of Z K by formulas: where n = dim K + 1. Note, that bigraded Betti numbers do not determine the dimension of K. The cone over K always has the same bigraded Betti numbers as K but the dimension is different. So far, β −i,2j (K) (together with dim K) is a very strong set of invariants. The question 1.4 makes sense for this set of invariants. Still the answer is negative.
Theorem 2. There exist simplicial complexes K 1 and K 2 such that We also show that Tor-algebras of the constructed complexes K 1 and K 2 have trivial multiplications. Thus not only bigraded Betti numbers but also multiplicative structure of H * (Z K ) does not determine Buchstaber invariant.
The paper consists of two essential parts which are independent from each other. Sections 2 and 3 form the first part. Theorem 1 is proved in Section 3. Section 2 clarifies the combinatorial meaning of Buchstaber invariants and contains definitions and constructions necessary for understanding the proof. In the second part of the paper we explore the connection between Buchstaber invariants and bigraded Betti numbers. This requires some basic homological algebra and the construction of the Taylor resolution of a Stanley-Reisner ring. Section 4 contains all necessary definitions and the proof of Theorem 2.
2. Combinatorial approach to Buchstaber invariants 2.1. Characteristic functions. A subgroup G ⊆ T m acts freely on a moment-angle complex Z K if and only if G intersects stabilizers of the action T m Z K trivially.
Lemma 2.1. Stabilizers of the action T m Z K are coordinate subtori T I ⊆ T m , corresponding to simplices I ∈ K.
Proof. The subgroup T I preserves the point In this section we suppose for simplicity that K does not have ghost vertices. In other words, {i} ∈ K for any i ∈ [m]. Let G ⊂ T m be a toric subgroup of rank s acting freely on Z K . Consider the quotient map φ : T m → T m /G, and fix an arbitrary coordinate representation T m /G ∼ = T r , where r = m − s. We get a map φ : T m → T r such that the restriction φ| T I to any stabilizer subgroup is injective. For each vertex i ∈ [m] consider an i-th coor- . . , λ r i ), called characteristic map (corresponding to the subgroup G ⊆ T m ). Since φ| T I is injective the characteristic map satisfies the condition:  Remark 2.3. Note that actually there is no 1-to-1 correspondence between freely acting subgroups and characteristic functions. The first reason is a choice of an isomorphism T m /G ∼ = T r which was arbitrary. The second reason is that characteristic function was defined only up to sign. Integral vectors (λ 1 i , λ 2 i , . . . , λ r i ) and −(λ 1 i , λ 2 i , . . . , λ r i ) determine the same 1-dimensional toric subgroup.

Generalized chromatic invariants.
Let K and L be simplicial complexes on sets V (K) and V (L), possibly infinite. A map f : V (K) → V (L) is called a simplicial map (or a map of simplicial complexes) if I ∈ K implies f (I) ∈ L. For a simplicial map we write f : K → L. A map f : K → L is called non-degenerate if |f (I)| = |I| for each simplex I ∈ K. The following general definition is due to R.Živaljević [28, def. 4.11].
Definition 2.4 (Generalized chromatic invariant). Let F = {T α | α ∈ A} be a family of "test" simplicial complexes and let wt : A → R be a real-valued function. A T α -coloring of K is just a non-degenerate simplicial map f : K → T α and γ (F ,wt) , the (F, wt)-chromatic number of K, is defined as the infimum of all weights over all T α -colorings,   Example 2.7. Many classical and new invariants in graph theory are generalized chromatic invariants. These include fractional and circular chromatic numbers [23], orthogonal colorings [19], quantum chromatic number [8].
Example 2.8. An integral vector v ∈ Z n , v = 0 is called primitive if v is not divisible by natural numbers other than 1. A collection of integral vectors I = {v 1 , . . . , v k } ⊂ Z n is called unimodular if I is a part of some basis of a lattice Z n . Clearly, any vector in a unimodular collection is primitive. A subcollection of a unimodular collection is unimodular.
Consider the simplicial complex U n in which: (1) vertices are primitive vectors of Z n ; (2) simplices are unimodular collections of vectors. Obviously, maximal simplices are bases of the lattice Z n , so dim U n = n − 1. Define the test family Similarly, define R U n as a simplicial complex on the set Z n 2 \ {0} in which I is a simplex if I is a set of binary vectors linearly independent over We can always assume that test families satisfy γ (F ,wt) (T α ) = wt(T α ) in Definition 2.4. This holds for the families described above.
Generalized chromatic invariants share a common property. If there exists a nondegenerate map g : K → L, then γ (F ,wt) (K) γ (F ,wt) (L). This fact follows easily from the definition: if f : L → F α is an F α -coloring of L, then f • g : K → L → F α is an F α -coloring of K with the same weight. For Buchstaber invariants (Example 2.8) this observation gives where m K , m L are the numbers of vertices of K and L. This fact was first pointed out by N.Erokhovets in [11].
On the other hand, the aforementioned monotonicity property is in general not substantial due of the following "general nonsense" argument. Claim 2.9. Let a(·) be an invariant of simplicial complexes taking values in R and such that a(K) a(L) if there exists a non-degenerate map g : K → L. Then a(·) is a generalized chromatic invariant.
Proof. Just take the family of all simplicial complexes weighted by a(·) itself. Of course, we suppose that all complexes under consideration belong to some "good universe" to avoid set-theoretical problems.
Let us describe the relation between different generalized chromatic invariants. Let (F 1 , wt 1 ) and (F 2 , wt 2 ) be weighted test families. We say that there is a morphism Ψ : (F 1 , wt 1 ) → (F 2 , wt 2 ) if for each complex T ∈ F 1 there exists a non-degenerate simplicial map from T to some S ∈ F 2 with wt 2 (S) wt 1 (T ).
The proof is immediate.
Lemma 2.11. There is a series of morphisms: for the families defined in Examples 2.5,2.6 and 2.8 Proof. Indeed, for each n ∈ N we have the following. (1) A non-degenerate map ∆ [n] → U n , sending [n] to a basis of the lattice Z n . (2) A non-degenerate map p : U n → R U n , which reduces each primitive vector (a vertex of U n ) modulo 2. The map p, obviously, sends unimodular collections from Z n to linearly independent sets in Z n 2 .
(3) A non-degenerate inclusion map R U n → ∆ n−1 ∞ . From Lemmas 2.10 and 2.11 follows for any K. Equivalently: The estimation of s(K) by ordinary chromatic number was first proved in [21]. The inequality between real and ordinary Buchstaber invariant can be understood topologically as well [16].
We call two test families equivalent, . Equivalent families define equal generalized chromatic invariants by Lemma 2.10. Therefore to prove that certain generalized chromatic invariants are different we need to prove that their test families are not equivalent.
In particular, to prove that r(·) and r R (·) are different invariants, it is sufficient to show that for some n ∈ N there is no non-degenerate map from R U n to U n . In other words, we should prove that r( R U n ) > n = r R ( R U n ) for some n ∈ N. This consideration is summarized as follows: Claim 2.12. If there exists a simplicial complex K such that s(K) = s R (K) then such complex can be found among { R U n | n ∈ N}.
We start to check complexes R U n for small values of n. For a test family (F, wt) define (F ( ) , wt) as a family of -skeletons of members of F.
Proof. For complexes K of dimension 0, 1, 2 a non-degenerate map from K to T is the same as a non-degenerate map from K to T (2) . Therefore, We prove that (F The proof exploits a trick invented in [12,14]. The modulo 2 reduction map p : U n is already constructed. Let us construct a non-degenerate map q : . It can be written as an array of 0 and 1. Consider q(v) ∈ Z n the same array of 0 and 1 as an integral vector. It is easily shown that if I ⊂ Z n 2 is a set of at most 3 linearly independent vectors, then The proposition now follows from Statement 2.2.
More can be said in the case dim K 1.
Here m is the number of vertices of K, γ(K) chromatic number, and [ · ] denotes an integral part.
, since both complexes are complete graphs on 2 n − 1 vertices. Thus the family (F Corollary 2.15. For finite 1-dimensional simplicial complexes (i.e. simple graphs) the problem to decide, whether s(K) (or s R (K)) is equal to m − 2, is NP-complete.

Real and ordinary Buchstaber invariants are different
In this section we prove Theorem 1, by showing that s( R U 4 ) > 4 = s R ( R U 4 ) for the complex R U 4 defined in the previous section. In other words, we prove that there is no non-degenerate simplicial map from R U 4 to U 4 .
Let e denote the nonzero element of Z 2 to avoid confusion with integral unit. Recall the map p : Lemma 3.1. Let f : R U 4 → N be a non-degenerate map. Then f is an injective map of vertices.
Proof. Vertices of R U 4 are pairwise connected. By non-degeneracy, Remark 3.2. Every non-degenerate map f : R U 4 → N is injective on simplices as well.
Lemma 3.3. If there exists a non-degenerate mapν : R U 4 → U 4 , then there exists a non-degenerate map ν : R U 4 → U 4 such that The map q is a non-degenerate simplicial map, therefore, by Lemma 3.1, it is injective on vertices of R U 4 . Thus q defines a permutation on a finite set of vertices V ( R U 4 ). Then q n = id for some n 1. Take ν =ν •q n−1 : R U 4 → U 4 . Then ν is a non-degenerate simplicial map, and p•ν = q n = id.
A non-degenerate map ν : R U 4 → U 4 will be called a lift if it satisfies (3.1). To prove the theorem it is sufficient to prove that lifts do not exist.
Suppose the contrary. Let Λ : R U 4 → U 4 be a lift. Vertices of R U 4 are, by definition, nonzero vectors of Z 4 2 . We list them in (3.2). Vectors at the right hand side of (3.2) are the values of Λ. Each vector at the right is a primitive vector in Z 4 . Since Λ is a lift, numbers a i are odd and b i are even. Values of Λ should satisfy ( * )-condition. It is reformulated for this particular case as follows: This is clear. Proof. Indeed, det(v 1 , v 2 , v 3 , v 4 ) = e, therefore, by ( * )-condition, Λ(v i ) i=1,2,3,4 is a basis of the lattice Z 4 . Expand all vectors Λ(v i ) in this basis.
Proof. Consider the matrix A over Z 2 :     The task is split into two steps to reduce the time of computation. GAP system [17] was used to perform this calculation. The implementation of described algorithm shows that there are no values of a i and b i for which ( * )-condition is satisfied. Thus r( R U 4 ) > 4 and which was to be proved.

Erokhovets criterium. A subset I ⊆ [m]
is called a minimal non-simplex (or a missing face) of K if I / ∈ K, but J ∈ K for any J I. The set of all minimal non-simplices of K is denoted by N (K).  [13,14]). The following conditions are equivalent: (1) s(K) 2; (2) s R (K) 2; (3) there exist J 1 , J 2 , J 3 ∈ N (K) such that J 1 ∩ J 2 ∩ J 3 = ∅. Sets J i are allowed to coincide.
The next example will be used in the proof of Theorem 2.
On the other hand there exist A 1 , A 2 , A 3 ∈ C 2 such that A 1 ∪ A 2 ∪ A 3 = S 9 . Now consider simplicial complexes L 1 and L 2 with N (L i ) = {I ⊂ S 9 | S 9 \ I ∈ C i } for i = 1, 2. Then condition (3) of Statement 4.1 does not hold for L 1 , but holds for L 2 . Statement 4.1 implies s(L 1 ) = 1 and s(L 2 ) 2 (and the same for s R ).
Both k and k[K] carry the structure of (multi)graded k (2) apply the functor ⊗ k[m] k; (3) take cohomology of the resulting complex: The resulting vector space inherits inner (multi)grading from R. It also obtains an additional grading − . It is well known that Tor * , * , k) does not depend on the choice of a free (multi)graded resolution R. Define bigraded Betti numbers of K as Several explicit constructions of free resolutions of k[K] are known. In our considerations we use one of the most important and basic constructions the Taylor resolution. In general, Taylor resolution is defined for any monomial ideal (see [25] or [24]). Here we concentrate only on the case of Stanley-Reisner rings, i.e. the case of square-free monomial ideals. The work [27] is also concerned with this particular case.
In the sequel the following convention is used. Any subset B ⊆ [m] determines the vector δ B ∈ Z m with i-th coordinate equal to 1 if i ∈ B and 0 otherwise. We simply write B ∈ Z m meaning δ B ∈ Z m . For a set B we denote the monomial v δ B = v Construction 4.6 (Taylor resolution). Consider the set N (K) of minimal non-simplices. Fix a linear order on N (K). For each J ∈ N (K) associate a formal variable w J and construct the free k[m]-module Here Λ [{w J }] is the vector space over k, generated by formal expressions W σ = w J1 ∧. . .∧w J for all subsets σ = {J 1 , . . . , J } ⊆ N (K), J 1 < . . . < J .
Define the multigrading and the double grading The first component is called a homological grading.
where v X σ,J i ∈ k[m] is the monomial corresponding to the set Define the multiplication on the k[m]-module R T = R − T by describing the products of generators. Let σ = {J 1 < . . . < J }, τ = {I 1 < . . . < I k } ⊆ N (K).
(1) The vector space R T = R − T is a differential Z m+1 -graded algebra over the ring k[m] w.r.t. to the multigrading, the differential, and the multiplication described above. This algebra is skew-commutative with respect to homological grading.     with the standard Grassmann product, bideg(w i ) = (−1, 2|M i |), and differential:    Proof. By definition, and J ∈ σ. This is equivalent to X σ,J = ∅. By definition, X σ,J = J \ I∈σ,I =J I . If the Taylor resolution is minimal, then, in particular, X N (K),J = ∅, which is precisely the condition (4.6) of the lemma. On the other hand, X N (K),J = ∅ implies X σ,J = ∅ for any σ ⊆ N (K). , k) has the following description: • It is generated as a vector space over k by W σ for σ ⊆ N (K); • The multidegree is given by (4.1); • The multiplication is given by The proof follows easily from the construction of Taylor resolution and the definition of minimality.
For complexes with the minimal Taylor resolution bigraded Betti numbers are expressed in combinatorial terms. At last, we have all necessary ingredients to prove Theorem 2. As a starting point take complexes L 1 and L 2 defined in Example 4.2. Our plan is the following: (1) To upgrade L 1 and L 2 to the new complexes K 1 and K 2 satisfying condition (4.6) (Taylor resolution is minimal); (2) To prove that β − ,2j (K 1 ) = β − ,2j (K 2 ) using formula (4.8); (3) To prove that s(K 1 ) = 1 and s(K 2 ) 2.
(4) Final technical remarks: dim(K 1 ) = dim(K 2 ), γ(K 1 ) = γ(K 2 ), and algebra iso- Step 1. Let L be any complex on a set [m] with the set of minimal non-simplices N (L). For each J ∈ N (L) consider a symbol a J . Define the complex L on the set V = [m] {a J | J ∈ N (K)} with the set of minimal non-simplices given by The Taylor resolution of the complex L is minimal. Indeed, any J ∈ N ( L) contains the vertex a J which does not belong to other minimal non-simplices. Therefore, condition (4.6) holds for L. Now we apply this construction to L 1 and L 2 . Recall that N (L i ) = {I ⊂ S 9 | S 9 \I ∈ C i } for i = 1, 2 and collections of subsets C 1 , C 2 shown on fig.1. Set K i = L i for i = 1, 2. Both K 1 and K 2 have 9 + 6 = 15 vertices.
Step 2. Apply (4.8) to K i : The last equality is the consequence of bijective correspondence between N (L i ) and N (K i ), sending J ∈ N (L i ) to J ∈ N (K i ). We have Returning to (4.10), The last equality follows from the definition of L i , since N (L i ) consists of complements to subsets of the collection C i . By analyzing fig.1 we see that for each and j Indeed, in both C 1 and C 2 there are 3 subsets of cardinality 2, 3 subsets of cardinality 3, 6 pairwise intersections of cardinality 1, and all other intersections are empty. Therefore, β − ,2j (K 1 ) = β − ,2j (K 2 ). The nonzero bigraded Betti numbers calculated by the described method are presented in fig.2 (empty cells are filled with zeroes).  Step 3. We use the following simple observation. Condition (3) of Statement 4.1 holds for the complex L whenever it holds for L. Indeed, As observed in Example 4.2 condition (3) of Statement 4.1 holds for L 2 and does not hold for L 1 . Therefore it also holds for K 2 = L 2 and does not hold for K 1 = L 1 . Thus s(K 1 ) = s(K 2 ) and s R (K 1 ) = s R (K 2 ).
Step 4. Final remarks. Suppose that S / ∈ K 1 . Then there exists J ∈ N (K 1 ) such that J ⊆ S. Therefore, {1, 4} = S 9 \ S ⊆ S 9 \ J. By construction, S 9 \ J ∈ C 1 . But {1, 4} is not contained in any A ∈ C 1 the contradiction. Thus S ∈ K 1 and dim K 1 |S| − 1 = 12. Similar reasoning shows that there is no simplex in K 1 of cardinality 14 (because any singleton lies in some A ∈ C 1 ). Therefore dim K 1 is exactly 12. Same for K 2 .
Remark 4.17. In both complexes K 1 and K 2 there are no minimal non-simplices of cardinality 1 and 2. Therefore all pairs of vertices in K 1 and K 2 are connected by edges, so 2 are complete graphs on 15 vertices. Thus chromatic numbers coincide γ(K 1 ) = γ(K 2 ) = 15.
Remark 4.18. Tor-algebras of K 1 and K 2 are isomorphic as algebras. Actually, the products in Tor k [15] (k[K 1 ], k) and Tor k [15] (k[K 2 ], k) are trivial by dimensional reasons (see fig. 2): products of nonzero elements always hit zero cells. The triviality of multiplication can be deduced also from Lemma 4.15 but this approach requires a complicated combinatorial reasoning.
These remarks conclude the proof of Theorem 2.  Remark 4.20. Question 1.4 may be formulated for an equivariant cohomology ring of Z K . This task is not interesting as well. Indeed, H * T m (Z K ; k) ∼ = k[K] (see [9] or [5]). It is known, that the Stanley-Reisner algebra k[K] determines the combinatorics of K uniquely [4]. Therefore multiplicative isomorphism H * T m (Z K1 ; k) ∼ = H * T m (Z K2 ; k) implies K 1 ∼ = K 2 and, in particular, s(K 1 ) = s(K 2 ).

Conclusion and open questions
Constructions of Buchstaber invariants and bigraded Betti numbers are defined for any simplicial complex. Nevertheless, in toric topology the most important are simplicial complexes arising from polytopes.
Let P be a simple polytope with m vertices. The polar dual polytope P * is simplicial. The complex K P = ∂P * is a simplicial sphere with m vertices. It is known [5,6] that Z K P is a smooth compact manifold and the action of T m on Z K P is smooth. The algebraic version of this fact is Avramov-Golod theorem [7,Th.3.4.4]. It states the following. The Tor-algebra Tor * , * k[m] (k[K]; k) is a (multigraded) Poincare duality algebra if and only if the complex K is Gorenstein*. Any simplicial sphere K is Gorenstein* [26, Th.5.1]. In particular, for any simple polytope P the complex K P is Gorenstein*, thus Tor * , * k[m] (k[K P ]; k) is a Poincare duality algebra. This is not surprising since Tor * , * k[m] (k[K P ]; k) ∼ = H * (Z K P ; k) and Z K P is a manifold.
The problems solved in this paper can be posed for particular classes of simplicial complexes, for example boundaries of simplicial polytopes or simplicial spheres. Problem 1. Does s(K P ) = s R (K P ) for any simple polytope P ?
The complex U = R U 4 constructed in the proof of Theorem 1 is not a boundary of a polytope; it is not a simplicial sphere as well. Nevertheless, U is Cohen-Macaulay as proved in [9,Th.2.2].
Another problem can also be formulated for the class of polytopes.
The complexes K 1 and K 2 constructed in Section 4 are not simplicial spheres as well. One can deduce this from the table of bigraded Betti numbers ( fig. 2): if the complexes were spheres the distribution of bigraded Betti numbers would be symmetric according to (bigraded) Poincare duality.
It is tempting to modify the construction of K 1 and K 2 of Section 4 to obtain polytopal spheres in the output. Unfortunately, this attempt fails due to the following observation.