Composition of simplicial complexes, polytopes and multigraded Betti numbers 3 1 Ayzenberg Anton 0 2 n Abstract. ForasimplicialcomplexKonmverticesandsimplicialcomplexesK1,...,Km a a composed simplicial complex K(K1,...,Km) is introduced. This construction gener- J alizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. 8 R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of 1 polytopes P(P1,...,Pm) defined by G. Agnarsson in most important cases. The com- positiondefinesastructureofanoperadonasetoffinitesimplicialcomplexes,inwhich ] acomplexonmverticesisviewedasanm-adicoperation. Weprovethefollowing: (1) O a composed complex K(K1,...,Km) is a simplicial sphere iff K is a simplicial sphere C andKi aretheboundariesofsimplices;(2)aclassofsphericalnerve-complexesisclosed under the operation of composition (3) finally, we express multigraded Betti numbers h. of K(K1,...,Km) in terms of multigraded Betti numbers of K,K1,...,Km using a t compositionofgeneratingfunctions. a m [ 1 v 9 1. Introduction 5 4 In toric topology multiple connections between convex polytopes, simplicial complexes, 4 topologicalspacesandStanley–Reisneralgebrasarestudied. Startingwithasimplepolytope . 1 P one constructs a moment-angle manifold Z with a torus action such that its orbit space P 0 is the polytope P itself. On the other hand, a simplicial complex ∂P∗ gives rise to a 3 moment-angle complex Z (D2,S1). This complex is homeomorphic to Z and possesses 1 ∂P∗ P a natural cellular structure which allows to describe its cohomology ring: H∗(Z ;k) ∼= : P v Tor∗,∗ (k[∂P∗],k). This consideration can be used to translate topological problems to i k[m] X the language of Stanley–Reisner algebras and vice-versa. Moreover, the cohomology ring r H∗(ZP;k)carriesaninformationaboutthecombinatoricsofthepolytopeP fromwhichwe a started. With some modifications this setting can be generalized to nonsimple polytopes. If P is a convex polytope (possibly nonsimple), then the moment-angle space Z is defined as P an intersection of real quadrics (but in nonsimple case Z is not a manifold). A simplicial P complexK ,calledthenerve-complex[3],isassociatedtoeachpolytope(innonsimplecase P K is not a simplicial sphere). The complex K carries a complete information on the P P combinatorics of P and its properties are similar to simplicial spheres. Generally there is a homotopy equivalence Z (cid:39) Z (D2,S1). An open question is to describe the properties ofStanley–ReisneralgebrPask[KKP]andcohomologyringsH∗(Z ;k)∼=Tor∗,∗ (k[K ],k)for P P k[m] P nonsimple convex polytopes. 1 2 AYZENBERGANTON In the work of A.Bahri, M.Bendersky, F.R.Cohen and S.Gitler [5] a new construc- tion is described, which allows to build a simple polytope P(l ,...,l ) from a given sim- 1 m ple polytope P with m facets and an array (l ,...,l ) of natural numbers. A simpli- 1 m cial complex ∂P(l ,...,l )∗ can be described combinatorially in terms of missing faces. 1 m Such description gives a representation of Z (D2,S1) as a polyhedral product ∂P(l1,...,lm)∗ ZP∗((D2li,S2li−1)) which leads, in particular, to alternative representation of the cohomol- ogy ring H∗(Z ). P(l1,...,lm) Theideaoftreatingnonsimplepolytopescanbeusedtocaptureawiderclassofexamples and find more general constructions. One of the constructions is known in convex geometry (we refer to the work of Geir Agnarsson [1]). Given a polytope P ⊂ Rm and polytopes (cid:62) P ,...,P a new polytope P(P ,...,P ) is constructed. This polytope generally depends 1 m 1 m on geometrical representation of P ⊂Rm, but under some restrictions the construction can (cid:62) be made combinatorial. In particular cases this construction gives the iterated polytope P(l ,...,l ) from the work [5]. Note, that the polytope P(P ,...,P ) may be nonsimple 1 m 1 m even in the case when all the polytopes P,P ,...,P are simple. 1 m In this work we introduce a new operation on the set of abstract simplicial com- plexes K,K ,...,K (cid:55)→ K(K ,...,K ). This operation corresponds to the operation 1 m 1 m P(P ,...,P ) on convex polytopes and generalizes the constructions of [5]. The work is 1 m organized as follows: (1) We review the construction of K and the definition of abstract spherical nerve- P complex from the work [3]. Section 2. (2) The construction of P(P ,...,P ). We give a few equivalent descriptions of this 1 m polytope and specialize the conditions under which P(P ,...,P ) is well defined 1 m on combinatorial polytopes. Section 3. (3) Given a simplicial complex K on m vertices and simplicial complexes K ,...,K 1 m we define a composed simplicial complex K(K ,...,K ), which is a central ob- 1 m ject of the work. Two equivalent definitions are provided: one is combinatorial, anotherdescribesK(K ,...,K )asananalogueofpolyhedralproductcalledpoly- 1 m hedral join. It is shown that K(∂∆ ,...,∂∆ ) = K(l ,...,l ) — an iterated [l1] [lm] 1 m simplicial wedge construction from the work [5]. We prove that K = P(P1,...,Pm) K (K ,...,K ). Section 4. P P1 Pm (4) Polyhedral products defined by composed simplicial complexes. In section 5 we review and generalize some results from [5]. (5) In section 6 the structure of composed simplicial complexes is studied. At first we describethehomotopytypeofK(K ,...,K ). IthappensthatK(K ,...,K )(cid:39) 1 m 1 m K ∗K ∗...∗K . The problem: for which choice of K,K ,...,K the complex 1 m 1 m K(K ,...,K ) is a sphere? The answer: only in the case, when K is a sphere 1 m and K = ∂∆ . Thus the class of simplicial spheres is not closed under the i [li] composition. Nevertheless, if K,K ,...,K are spherical nerve-complexes, then 1 m so is K(K ,...,K ). 1 m (6) Insection7wedescribethemultigradedBettinumbersofK(K ,...,K ). Thereis 1 m asimpleformulawhichexpressesthesenumbersintermsofmultigradedBettinum- bers of K,K ,...,K . Applying this formula to ∂∆ (K ,K ) and o2(K ,K ), 1 m [2] 1 2 1 2 whereo2 isthecomplexwith2ghostvertices,givestheresultof[3]. Usingthecon- nection between bigraded Betti numbers and h-polynomial, found by V.M.Buch- staber and T.E.Panov [7], in section 8 we provide formulas for h-polynomials of compositions in some particular cases. Some of these formulas were found earlier by Yu.Ustonovsky [16]. COMPOSITION OF SIMPLICIAL COMPLEXES, POLYTOPES AND MULTIGRADED BETTI NUMBERS 3 The following notation and conventions are used. The simplicial complex K on a set of vertices [m] is the system of subsets K ⊆ 2[m], such that I ∈ K and J ⊂ I implies J ∈ K. A vertex i ∈ [m] such that {i} ∈/ K is called ghost vertex. If I ∈ K, then link I is the K simplicialcomplexonaset[m]\I suchthatJ ∈link I ⇔J(cid:116)I ∈K. Notethatalinkmay K have ghost vertices even if K does not have them. From the geometrical point of view the complex does not change when ghost vertices are omitted. We use the same symbol for the simplicial complex K and its geometrical realization. The complex K is called a simplicial sphere if it is PL-homeomorphic to the boundary of a simplex (we omit ghost vertices if necessary). SimplicialcomplexK iscalledageneralizedhomologicalsphere(orGorenstein* complex) if K and all its links have homology of spheres of corresponding dimensions. If K is a simplicial sphere (resp. Gorenstein* complex) then so is link I for each I ∈K. K If A⊂[m], then full subcomplex K is the complex on A such that J ∈K ⇔J ∈K. A A We denote the full simplex on the set [m] by ∆ , it has dimension m−1. Its boundary [m] ∂∆ — is complex on [m], consisting of all proper subsets of [m]. [m] The notation x¯ = (x ,...,x ) ∈ Rm is used for arrays of numbers, and (cid:104)x¯,y¯(cid:105) denotes 1 m thesumx y +x y +...+x y . Sometimesdoublearrayswillbeused: x¯=(x¯ ,...,x¯ )= 1 1 2 2 m m 1 m (x ,...,x ,...,x ,...,x ). 11 1l1 m1 mlm I wish to thank Anthony Bahri for the private discussion in which he explained the geometrical meaning of the simplicial wedge construction and for his comments on the subject of this work. I am also grateful to Nickolai Erokhovets for paying my attention to the work of Geir Agnarsson [1]. 2. Polytopes and nerve-complexes Let P be an n-dimensional polytope and let {F ,...,F } be the set of all its facets. 1 m Consider a simplicial complex K on the set [m] = {1,...,m} called the nerve-complex of P a polytope P, defined by the condition I ={i ,...,i }∈K whenever F ∩...∩F (cid:54)=∅. 1 k P i1 ik The complex K is thus the nerve of the closed covering of the boundary ∂P by facets. P Example 2.1. If P is simple, then K coincides with a boundary of a dual simplicial P polytope: K = ∂P∗. In this case K is a simplicial sphere. It can be shown that K is P P P not a sphere if P is not simple. As shown in [3] nerve-complexes are nice substitutes for nonsimple polytopes. In particu- lar, the moment-angle space Z of any convex polytope P is homotopy equivalent to the P moment-angle complex Z (D2,S1), the Buchstaber numbers s(P) and s(K ) are equal, KP P etc. TherearenecessaryconditionsonthecomplexKtobethenerve-complexofsomeconvex polytope. These conditions are gathered in the notion of a spherical nerve-complex. Let K be a simplicial complex, M(K) — the set of its maximal (under inclusion) sim- plices. Let F(K) = {I ∈ K | I = ∩J , where J ∈ M(K)}. The set F(K) is partially i i ordered by inclusion. It can be shown (see [3]) that for each simplex I ∈/ F(K) the complex link I is contractible. K Definition 2.2 (Spherical nerve-complex). Simplicial complex K is called a spherical nerve-complex of rank n if the following conditions hold: • ∅∈F(K), i.e. intersection of all maximal simplices of K is empty; 4 AYZENBERGANTON • F(K) is a graded poset of rank n (it means that all its saturated chains have the cardinality n+1). In this case the rank function rank: F(K) → Z(cid:62) is defined, such that rank(I)= the cardinality of saturated chain from ∅ to I minus 1. • For any simplex I ∈ F(K) the simplicial complex link I is homotopy equivalent K to a sphere Sn−rank(I)−1. Here, by definition, link ∅=K and S−1 =∅. K Statement 2.3. If P is an n-dimensional polytope, then K is a spherical nerve- P complex of rank n and, moreover, the poset F(K ) is isomorphic to the poset of faces of P P ordered by reverse inclusion. As a corollary, the poset of faces of P can be restored from K , thus K is a complete P P invariant of a combinatorial polytope P. 3. Composition of polytopes Let[m]={1,...,m}beafinitesetand(cid:52) beastandard(m−1)-dimensionalsimplex [m] in Rm given by {x¯=(x ,...,x )∈Rm |x (cid:62)0;(cid:80)x =1}. The convex polytope P ⊂Rm 1 m i i will be called stochastic if P ⊆(cid:52) . The following definition is due to [1, def.4.5]. [m] Definition 3.1. Let P ⊆ Rm and Pi ⊆ Rli for i ∈ [m] be stochastic polytopes. The polytope (3.1) P(P ,...,P )={(t x¯ ,t x¯ ,...,t x¯ )∈R(cid:80)li | 1 m 1 1 2 2 m m |t¯=(t ,...,t )∈P,x¯ ∈P for each i} 1 m i i is called the composition of polytopes P and {P }. i In [1] this operation is called the action of P. Example 3.2. (cid:52) (P ,...,P )=P ∗...∗P — the join of polytopes. [m] 1 m 1 m Theoriginalmotivationofdefinition3.1wastoextendthenotionofthejointomoregeneral convex sets of parameters t . i Remark 3.3. Definition 3.1 depends crucially on the geometrical representation of poly- topes, not only their combinatorial type. Definition 3.4. LetL⊆Rm beanaffinen-dimensionalsubspacesuchthatP =L∩Rm (cid:62) is a nonempty bounded set (thus a polytope). If P is a stochastic polytope and every facet F ⊂ P is defined uniquely as F = P ∩ {x = 0} we call P a natural (stochastic) i i i polytope. Remark 3.5. A natural stochastic polytope P in Rm has exactly m facets. For a point x¯∈Rm define σˆ(x¯)={i∈[m]|x =0}. (cid:62) i Remark 3.6. ForanaturalstochasticpolytopeP ⊆Rm thenerve-complexcanbedefined (cid:62) by the condition: I ∈K , whenever there exists a point x¯∈P such that I ⊆σˆ(x¯). Indeed, P I ∈ K implies that (cid:84) F (cid:54)= ∅. Let x¯ ∈ (cid:84) F . Then x = 0 for each i ∈ I therefore P i∈I i i∈I i i I ∈σˆ(x¯). Observation 3.7. Any polytope P is affine equivalent to a natural stochastic polytope. COMPOSITION OF SIMPLICIAL COMPLEXES, POLYTOPES AND MULTIGRADED BETTI NUMBERS 5 Observation 3.8. The space L⊆Rm in the definition 3.4 can be defined by the system of affine relations L = {x¯ ∈ Rm | (cid:80) cjx +d = 0 for i = 1,...,m−n} where all the j i j i coefficients cj are positive and d =−1. i i Proof of both observations. Let (3.2) P ={y¯∈Rn |(cid:104)a¯ ,y¯(cid:105)+b (cid:62)0,i∈[m]} i i bearepresentationofP asanintersectionofhalfspaces,wherea¯ istheinnernormalvector i to the i-th facet (we suppose that there are no excess inequalities in (3.2) and |a¯ |=1). i Consideranaffineembeddingj : Rn →Rm,givenbyj (y¯)=((cid:104)a¯ ,y¯(cid:105)+b ,...,(cid:104)a¯ ,y¯(cid:105)+ P P 1 1 m b ). Obviously, j (P) ⊆ Rm and, moreover, j (P) = j (Rn)∩Rm. Denote the affine m P (cid:62) P P (cid:62) subspace j (Rn) by L. This subspace is given by the system of affine relations L = {x¯ ∈ P Rm |(cid:104)c¯,x¯(cid:105)+d =0fori=1,...,m−n}. Thefacetsofj (P)aregivenbyj (P)∩{x =0}. i i P P i (cid:80) Notice that there is a relation S a¯ =0 by Minkowski theorem, where S >0 are the i i i (cid:80) (n−1)-volumesoffacets. ThenoneoftheaffinerelationsforLhastheform S x +d=0 i i i with all the coefficients S strictly positive. Adding this relation multiplied by large enough i number to other relations leads to a system of relations with positive coefficients. Now divide each relation by d to get the relations of the form (cid:80)cjx = 1. Set new i i j variables x(cid:48) = cjx to transform one of the relations to the form (cid:80)x = 1. This gives a j 1 j j stochastic polytope in Rm. Observations proved. (cid:3) Proposition 3.9. Let P ∈ Rm be a natural stochastic polytope given by P = Rm ∩ (cid:62) {(cid:104)c¯,x¯(cid:105) = 1,i = 1,...,m − n}, c¯ = (c1,...,cm) and for each i ∈ [m] a natural sto- i i i i chastic polytope Pi ∈ Rli is given by Pi = Rl(cid:62)i ∩ {(cid:104)c¯iji,x¯i(cid:105) = 1,ji = 1,...,li − ni}, c¯ = (c1 ,...,cli ). Then the polytope P(P ,...,P ) is a natural stochastic polytope de- iji iji iji 1 m scribed by the system (3.3) P(P ,...,P )={(x¯ ,...,x¯ )∈Rl1 ×...×Rlm =R(cid:80)li | 1 m 1 m (cid:62) (cid:62) (cid:62) |c1(cid:104)c¯ ,x¯ (cid:105)+c2(cid:104)c¯ ,x¯ (cid:105)+...+cm(cid:104)c¯ ,x¯ (cid:105)=1} i 1j1 1 i 2j2 2 i mjm m Proof. BydirectsubstitutionP(P ,...,P )asdefinedin3.1satisfiesallthespecified 1 m affine relations. On the contrary let x¯ = (x¯ ,...,x¯ ) ∈ R(cid:80)li satisfies relations (3.3) for 1 m (cid:62) all i,j ,...,j . Denote (cid:104)c¯ ,x¯ (cid:105) ∈ R by t (j ). Then t (j ) (cid:62) 0 (by nonnegativity of 1 m iji i i i i i coefficients in affine relations) and c1t (j ) + c2t (j ) + ... + cmt (j ) = 1 for each i, i 1 1 i 2 2 i m m therefore t¯(¯j)=(t (j ),...,t (j ))∈P. 1 1 m m Let us show that t (j ) does not actually depend on j . Consider first entry j for i i i 1 simplicity. Let j and j(cid:48) be different indices. The point x¯ satisfies the relations 1 1 c1(cid:104)c¯ ,x¯ (cid:105)+c2(cid:104)c¯ ,x¯ (cid:105)+...+cm(cid:104)c¯ ,x¯ (cid:105)=1 i 1j1 1 i 2j2 2 i mjm m and c1(cid:104)c¯ ,x¯ (cid:105)+c2(cid:104)c¯ ,x¯ (cid:105)+...+cm(cid:104)c¯ ,x¯ (cid:105)=1 i 1j1(cid:48) 1 i 2j2 2 i mjm m Subtracting we get c1t (j ) = c1(cid:104)c¯ ,x¯ (cid:105) = c1(cid:104)c¯ ,x¯ (cid:105) = c1t (j(cid:48)). Since c1 (cid:54)= 0 (at least i 1 1 i 1j1 1 i 1j1 1 i 1 1 i for one i) we get t (j )=t (j(cid:48)). 1 1 1 1 Thus far we can simply write t instead of t (j ). Then t¯= (t ,...,t ) ∈ P. As a i i i 1 m consequence, (cid:104)c¯iji,x¯tii(cid:105) = 1 for each i and ji. Then x¯ = (t1x¯t11,t2x¯t22,...,tmx¯tmm) where t¯∈ P and x¯tii ∈Pi. This means x¯∈P(P1,...,Pm) by definition. (cid:3) 6 AYZENBERGANTON Example 3.10. Let P ⊆Rm be a natural stochastic polytope (with m facets) defined by (cid:62) rPpeo.llayItttiooipnsseg{iPv(cid:104)e(c¯lni1,,x¯i.n(cid:105)..=R,(cid:80)l1m}l)ia=bnydPt∆h((cid:52)e[li[s]ly1⊆]s,t.eR.m.li,o(cid:52)afs[latmffia]n)nd⊆earrRdel(cid:80)asitmliiopnislsecxalglievdenthbeyit{exr1a+ti.o.n.+ofxtlihe=p1o}ly.tTophee (cid:62) (3.4) c1(x +...+x )+c2(x +...+x )+...+cm(x +...+x )=1. i 11 1l1 i 21 2l2 i m1 mlm If P is simple then P(l ,...,l ) is simple as well (see section 6 or the work [5]). Such 1 m polytopes, their quasitoric manifolds and moment-angle complexes were studied in [5]. For the particular case P(l,...,l), l>0 we use the notation lP. Remark3.11. Insection6wewillshowthatfornaturalstochasticpolytopestheoperation P(P ,...,P ) depends up to combinatorial equivalence only on the combinatorial type of 1 m polytopes. Since each polytope has a natural stochastic representation we can view the composition as the operation on combinatorial polytopes. Proposition 3.12 (Associativity law for the composition of polytopes). Let P be a stochastic polytope in Rm, P ,...,P be stochastic polytopes in Rl1,...,Rlm respectively (cid:62) 1 m (cid:62) (cid:62) and P ,...,P ,P ,...,P ,...,P ,...,P 11 1l1 21 2l2 m1 mlm — stochastic polytopes as well. Then (3.5) P(P (P ,...,P ),...,P (P ,...,P ))= 1 11 1l1 m m1 mlm P(P ,...,P )(P ,...,P ,...,P ,...,P ). 1 m 11 1l1 m1 mlm The proof follows easily from the definition 3.1. Remark 3.13. It can be seen that P(pt,...,pt)=pt(P)=P, where pt=(cid:52) is a point. [1] Thus far the set of all stochastic polytopes carries the structure of an operad, where the polytope in Rm is viewed as m-adic operation and the composition is given by the compo- (cid:62) sition of polytopes described above. Proposition 3.12 expresses the associativity condition for the operad and the polytope pt is the “identity” element. Natural stochastic polytopes form a suboperad by proposition 3.9. 4. Composition of simplicial complexes ConsiderasimplicialcomplexKonmverticesandasetoftopologicalpairs{(X ,A )} , i i i∈[m] A ⊆ X . For a simplex I ∈ K let V be the subset of X × ... × X given by V = i i I 1 m I C ×...×C , where C =X if i∈I and C =A if i∈/ I. The space 1 m i i i i (cid:91) (cid:89) Z ((X ,A ))= V ⊆ X K i i I i I∈K i is called the polyhedral product of pairs (X ,A ) defined by K. i i Example 4.1. The motivating examples of polyhedral products are moment-angle com- plexes Z (D2,S1), real moment-angle complexes Z (D1,S0) and Davis–Januszkiewicz K K spaces DJ(K) = Z (CP∞,pt) (see [8]). Another series of examples is given by wedges K (cid:87) X ∼=Z ((X ,pt)),fatwedgesZ ((X ,pt))andgeneralizedfatwedgesZ ((X ,pt)). α α ∆(0) α ∂∆[m] α ∆(k) α [m] [m] The spaces of the form Z ((X ,pt)) were studied in [2]. The most general situation K α COMPOSITION OF SIMPLICIAL COMPLEXES, POLYTOPES AND MULTIGRADED BETTI NUMBERS 7 Z ((X ,A )) was defined and studied by A. Bahri, M. Bendersky, F. R. Cohen and S. K i i Gitler in [4] from the homotopy point of view. Theverynaturalthingistosubstitutethetopologicalproductinthedefinitionofapolyhe- dralproductbyanyotheroperationontopologicalspaces. Thusfarwecangetpolyhedral smash product Z∧((X ,A )) [4] and polyhedral join Z∗((X ,A )) as defined below. K i i K i i Definition 4.2. Let {(X ,A )} be topological pairs and K a simplicial complex i i i∈[m] on [m]. For each simplex I ∈ K consider a subset V ⊆ X ∗...∗X of the form V = I 1 m I C ∗...∗C , where C =X if i∈I and C =A if i∈/ I. The space 1 m i i i i (cid:91) Z∗((X ,A ))= V ⊆(cid:62)X K i i I i I∈K is called the polyhedral join of pairs (X ,A ). i i Observation 4.3. If X is a simplicial complex and A its simplicial subcomplex, the i i space Z∗((X ,A )) has a canonical simplicial structure. So far the polyhedral join is well K i i defined on the category of simplicial complexes as opposed to polyhedral product. Let K be a simplicial complex on the set [m]. It can be considered as a subcomplex of ∆ — the simplex on the set [m], so far there is a pair (∆ ,K). [m] [m] Definition 4.4. Let K be a simplicial complex on the set [m] and K a simplicial com- i plexontheset[l ]foreachi∈[m]. ThesimplicialcomplexK(K ,...,K )=Z∗((∆ ,K )) i 1 m K [li] i will be called the composition of K with K ,i∈[m]. i Now we define the composition of simplicial complexes in purely combinatorial terms. Let K be a simplicial complex on m vertices, which are possibly ghost. Let K ,...,K be 1 m simplicialcomplexesonthesets[l ],...,[l ](ghostverticesareallowedaswell). ThenK(K ) 1 m i is a simplicial complex on the set [l ](cid:116)...(cid:116)[l ] defined by the following condition: the set 1 m I =I (cid:116)...(cid:116)I ,I ⊆[l ]isthesimplexofK(K ,...,K )whenever{i∈[m]|I ∈/ K }∈K. 1 m i i 1 m i i K 5 K K 6 K 4 K K 3 1 2 K 7 K 8 K K(K ,...,K ) 1 8 Figure 1. Vertices of K(K ,...,K ) 1 m 8 AYZENBERGANTON TheprocessofconstructingthecomplexK(K )=K(K ,...,K )isdepictedonfigures α 1 m 1 and 2. The set of vertices of K(K ) is the union of vertices of K , which can be depicted α i by a simple diagram (fig. 1). To construct the simplex of K(K ) we fix any simplex J ∈K α andtakefullsubcomplex∆ (oranyofitsfaces)fori∈J andanysimplexI ∈K foreach [li] i i i∈/ J. The union of these sets gives a simplex of K(K ) (fig. 2). All simplices I ∈K(K ) α α can be constructed by such procedure. This approach to the construction of K(K ) will be α discussed in section 6 in more detail. I 6 I I I 7 1 2 I 8 J K(K ,...,K ) 1 8 Figure 2. Simplex of K(K ,...,K ) 1 m Let ol be the simplicial complex on l >0 vertices which has no nonempty simplices. It means that all its vertices are ghost. We formally set ∂∆ =o1. [1] Byremark3.6wecansetKpt =o1 sincethepolytopept=(cid:52)[1] isdefinedbyR(cid:62)∩{x∈ R|x=1} and does not intersect the hyperplane {x=0}. Example 4.5. We have by definition K(o1,...,o1)=K and o1(K)=K. Example 4.6. The complex K(ol,o1,...,o1). Let v be the first vertex of K. Then 1 K(ol,o,...,o) can be described by the following procedure: the vertex v is replaced by 1 a simplex I = {v1,...,vl} and simplices I ∈ K, containing v are blown up to simplices 1 1 1 1 (I\{v })(cid:116)I . Therefore, K(ol,o,...,o)=K ∪(link v )∗I . 1 1 [m]\v1 K 1 1 Example 4.7. om(K ,...,K )=K ∗...∗K . 1 m 1 m Next statement provides a connection between the composition of polytopes (in the nat- ural stochastic case) and the composition of simplicial complexes. Proposition 4.8. Let P,P ,...,P be natural stochastic polytopes. Then 1 m K =K (K ,...,K ). P(P1,...,Pm) P P1 Pm Proof. Weneedatechnicallemma. Recallfromsection3thatforx¯∈Rm,t¯(x¯)={i∈ (cid:62) [m]|x =0}. i COMPOSITION OF SIMPLICIAL COMPLEXES, POLYTOPES AND MULTIGRADED BETTI NUMBERS 9 Lemma 4.9. Let Q be a polytope given by Rm ∩{(cid:80)cjx = 1,i = 1,...,m−n} with (cid:62) i j cj > 0. Fix y ∈ R. If x¯ ∈ Rm is the solution to the system of equations (cid:80)cjx = y, then i (cid:62) i j y (cid:62)0 and either σˆ(x¯)∈K if y >0 or x¯=¯0 if y =0. Q Proof. If y =0, the statement is evident since cj >0 and x¯ should be nonnegative. If i y > 0 consider the point x¯/y. It can be seen that x¯/y ∈ Q and σˆ(x¯/y) = σˆ(x¯). Therefore σˆ(x¯)∈K . (cid:3) Q Let x¯ = (x ,...,x ) ∈ Rl. If x¯ ∈ P, then σˆ(x¯) ∈ K . Vice-versa, if I ∈ K then there 1 l P P exists x¯∈P such that I ⊆σˆ(x¯) (remark 3.6). It can be seen that both complexes K and K (K ,...,K ) have the same P(P1,...,Pm) P P1 Pm set of vertices [l ](cid:116)...(cid:116)[l ]. Denote l +...+l by Σ. Let 1 m 1 m x¯=(x ,...,x ,x ,...,x ,...,x ,...,x )∈RΣ =(x¯ ,...,x¯ ) 11 1l1 21 2l2 m1 mlm (cid:62) 1 m be the point of P(P ,...,P ). Then for the point x¯ of P(P ,...,P ) we have 1 m 1 m (cid:104)c¯,((cid:104)c¯ ,x¯ (cid:105),(cid:104)c¯ ,x¯ (cid:105),...,(cid:104)c¯ ,x¯ (cid:105))(cid:105)=1. i 1j1 1 2j2 2 mjm m Denote (cid:104)c¯ ,x¯ (cid:105) by t (it does not depend on j ,...,j — see proof of proposition 3.9) and sjs s s 1 m set t¯= (t ,...,t ). By observation 3.8 we may assume t (cid:62) 0. Therefore, σˆ(t¯) ∈ K . For 1 m s P all s∈[m] we have an alternative: • If s∈σˆ(t¯), then t =0 and (cid:104)c¯ ,x¯ (cid:105)=0. Then x¯ =¯0 by lemma 4.9. s sjs s s • If s∈/ σˆ(t¯), then t (cid:54)=0 and (cid:104)c¯ ,x¯ (cid:105)=t >0. Then by lemma 4.9 σˆ(x¯ )∈K . s sjs s s s Ps Therefore,σˆ(x¯)∈K (K ,...,K ). PrecedingargumentsshowthatifI ∈K , P P1 Pm P(P1,...,Pm) then I ∈ K (K ,...,K ). Now let J ∈ K (K ,...,K ), J = A (cid:116)...(cid:116)A , where P P1 Pm P P1 Pm 1 m A ⊆[l ]. We need to show that there exists a point x¯∈P(P ,...,P ) such that J ∈σˆ(x¯). s s 1 m By definition there exists a simplex I ∈ K such that s ∈/ I implies A ∈ K . There P s Ps exists a point t¯=(t ,...,t )∈P such that I ⊆σˆ(t¯). Also for each s there exist solutions 1 m to the system of equations {(cid:104)c¯ ,x¯ (cid:105) = t } such that A ⊆ σˆ(x¯ ) if t (cid:54)= 0 and s,js s s js=1,...,ls s s s x¯ =¯0 if t =0 (equiv. σˆ(x¯ )=[l ]⊇A ). Then the nonnegative solution to the system of s s s s s equations (cid:104)c¯,((cid:104)c¯ ,x¯ (cid:105),(cid:104)c¯ ,x¯ (cid:105)),...,(cid:104)c¯ ,x¯ (cid:105))(cid:105)=1 i 1j1 1 2j2 2 mjm m is given by x¯=(x¯ ,x¯ ,...,x¯ ), where σˆ(x¯)=σˆ(x¯ )(cid:116)...(cid:116)σˆ(x¯ )⊇J. This concludes the 1 2 m 1 m proof. (cid:3) Corollary 4.10. If P,P ,...,P are combinatorially equivalent to Q,Q ,...,Q re- 1 m 1 m spectively and all the polytopes are natural stochastic, then P(P ,...,P ) is combinatorially 1 m equivalent to Q(Q ,...,Q ). Therefore, P(P ,...,P ) can be viewed as a well-defined op- 1 m 1 m eration on combinatorial polytopes. Example 4.11. A nontrivial example of the composition is the iterated simplicial wedge construction as defined in [5]. Let K be a simplicial complex on m vertices and (l ,...,l )—anarrayofnaturalnumbers. ConsiderthesimplicialcomplexK(l ,...,l )= 1 m 1 m K(∂∆ ,...,∂∆ ). [l1] [lm] IfP isapolytope,thenK (l ,...,l )=K (∂∆ ,...,∂∆ )=K (K ,...,K )= P 1 m P [l1] [lm] P (cid:52)[l1] (cid:52)[lm] K =K by proposition 4.8. In section 6 we will show that for every P((cid:52)[l1],...,(cid:52)[lm]) P(l1,...,lm) m-tuple (l ,...,l ) simplicial complex K is a combinatorial sphere whenever K(l ,...,l ) 1 m 1 m 10 AYZENBERGANTON is a combinatorial sphere. Then P is simple whenever P(l ,...,l ) is simple (see example 1 m 2.1). Proposition 4.12 (Associativity law for the composition of simplicial complexes). Let K be a simplicial complex on m vertices, K ,...,K be simplicial complexes on l ,...,l 1 m 1 m vertices respectively and K ,...,K ,K ,...,K ,...,K ,...,K — simplicial com- 11 1l1 21 2l2 m1 mlm plexes on sets [r ] of vertices. Then sjs (4.1) K(K (K ,...,K ),...,K (K ,...,K ))= 1 11 1l1 m m1 mlm K(K ,...,K )(K ,...,K ,...,K ,...,K ) 1 m 11 1l1 m1 mlm (cid:70) as the complexes on the set [r ]. s,js sjs Proof. Both complexes have the same set of vertices V = ([r ](cid:116)...(cid:116)[r ])(cid:116)...(cid:116) 11 1l1 ([r ](cid:116)...(cid:116)[r ])LetAbethesubsetofV soA=(A (cid:116)...(cid:116)A )(cid:116)...(cid:116)(A (cid:116)...(cid:116)A ), m1 mlm 11 1l1 m1 mlm where A ⊆[r ]. The chain of equivalent conditions is written below. sjs sjs (4.2) A∈K(K (K ,...,K ),...,K (K ,...,K ))⇔ 1 11 1l1 m m1 mlm ∃I ∈K∀s∈/ I: (A (cid:116)...(cid:116)A )∈K (K ,...,K )⇔ s1 sls s s1 sls ∃I ∈K∀s∈/ I∃I ∈K ∀i ∈/ I : A ∈K ⇔ s s s s sis sis ∃J ∈K(K ,...,K )∀s∀i ∈[l ]\J: A ∈K ⇔ 1 m s s sis sis A∈K(K ,...,K )(K ,...,K ,...,K ,...,K ). 1 m 11 1l1 m1 mlm This finishes the proof. (cid:3) Remark 4.13. As in the case of polytopes simplicial complexes form an operad. The simplicial complex K on m vertices can be viewed as an m-adic operation. The “identity operation” is given by the complex o1 (see example 4.6) since K(o1,...,o1)=o1(K)=K. Corollary 4.14. The composition can be constructed by steps. More precisely, let K i be the complex on [l ], then i K(K ,...,K )=K(o1,...,K ,...,o1)(K ,...,K ,o1,...,o1,K ,...,K ). 1 m i 1 i−1 i+1 m (cid:124) (cid:123)(cid:122) (cid:125) li Corollary 4.15 ([5, sect.2]). Let l be natural numbers. Then i K(l ,...,l )=K(1,...,l ,...,1)(l ,...,l ,1,...,1,l ,...,l ). 1 m i 1 i−1 i+1 m (cid:124) (cid:123)(cid:122) (cid:125) li Onecan“blowup” verticesstepbystep. TheoperationK(l ,1,1,...,1)canbedescribed 1 geometrically [5], [14]: K(l,1,1,...,1)=K ∗∂∆ ∪(link {1})∗∆ . [m]\{1} [l] K [l] The figure 3 illustrates the situation when K is the boundary of 5-gon and l=2. It can be directly checked that K(l,1,1,...,1) ∼= Σl−1K ∼= K ∗ ∂∆ . In the PL PL [l] case when K is the boundary of simplicial polytope the complex K(l,1,1,...,1) is also the boundary of a polytope [5, Th.2.3]. Indeed, if K = ∂Q, then K = K for dual simple Q∗ polytope Q∗, then K(l ,...,l )=K =∂(Q∗(l ,...,l ))∗. Then, using corollary 1 m Q∗(l1,...,lm) 1 m 4.15 inductively, we get the following.