A combinatorial proof of a formula for Betti numbers of a stacked polytope

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1 A combinatorial proof of a formula for Betti numbers of a staced polytope Suyoung Choi Department of Mathematical Sciences KAIST, Republic of Korea choisy@aistacr (Current Department of Mathematics Osaa City University, Japan choi@sciosaa-cuacjp Jang Soo Kim Department of Mathematical Sciences KAIST, Republic of Korea jsim@aistacr (Current LIAFA University of Paris 7, France Submitted: Aug 8, 2009; Accepted: Dec 3, 2009; Published: Jan 5, 200 Mathematics Subject Classifications: 05A5, 05E40, 05E45, 52B05 Abstract For a simplicial complex, the graded Betti number β i,j ([ ] of the Stanley- Reisner ring [ ] over a field has a combinatorial interpretation due to Hochster Terai and Hibi showed that if is the boundary complex of a d-dimensional staced polytope with n vertices for d 3, then β, ([ ] = ( ( n d We prove this combinatorially Introduction A simplicial complex on a finite set V is a collection of subsets of V satisfying if v V, then {v}, 2 if F and F F, then F Each element F is called a face of The dimension of F is defined by dim(f = F The dimension of is defined by dim( = max{dim(f : F } For a subset W V, let W denote the simplicial complex {F W : F } on W The research of the first author was carried out with the support of the Japanese Society for the Promotion of Science (JSPS grant no P09023 and the Brain Korea 2 Project, KAIST The second author was supported by the SRC program of Korea Science and Engineering Foundation (KOSEF grant funded by the Korea government (MEST (No R the electronic journal of combinatorics 7 (200, #R9

2 Let be a simplicial complex on V Two elements v, u V are said to be connected if there is a sequence of vertices v = u 0, u,,u r = u such that {u i, u i+ } for all i = 0,,, r A connected component C of is a maximal nonempty subset of V such that every two elements of C are connected Let V = {x, x 2,, x n } and let R be the polynomial ring [x,,x n ] over a fixed field Then R is a graded ring with the standard grading R = i 0 R i Let R( j = i 0 (R( j i be the graded module over R with (R( j i = R j+i The Stanley-Reisner ring [ ] of over is defined to be R/I, where I is the ideal of R generated by the monomials x i x i2 x ir such that {x i, x i2,,x ir } A finite free resolution of [ ] is an exact sequence 0 F r φ r F r φ r φ 2 F φ F 0 φ 0 [ ] 0, ( where F i = j 0 R( j β i,j and each φ i is degree-preserving A finite free resolution ( is minimal if each β i,j is smallest possible There is a minimal finite free resolution of [ ] and it is unique up to isomorphism If ( is minimal, then the (i, j-th graded Betti number β i,j ([ ] of [ ] is defined to be β i,j ([ ] = β i,j Hochster s theorem says β i,j ([ ] = W V W =j dim Hj i ( W ; We refer the reader to [, 5] for the details of Betti numbers and Hochster s theorem Since dim H0 ( W ; is the number of connected components of W minus, we can interpret β i,i ([ ] in a purely combinatorial way Definition Let be a simplicial complex on a finite nonempty set V Let be a nonnegative integer The -th special graded Betti number b ( of is defined to be b ( = W V W = (cc( W, (2 where cc( W denotes the number of connected components of W Note that since there is no connected component in = { }, we have b 0 ( = If > V, then b ( = 0 because there is nothing in the sum in (2 Thus we have { β, ([ ], if, b ( =, if = 0 We refer the reader to [7] for the basic notions of convex polytopes Let P be a simplicial polytope with vertex set V The boundary complex (P is the simplicial complex on V such that F for some F V if and only if F V and the convex hull of F is a face of P Note that if the dimension of P is d, then dim( (P = d the electronic journal of combinatorics 7 (200, #R9 2

3 For a d-dimensional simplicial polytope P, we can attach a d-dimensional simplex to a facet of P A staced polytope is a simplicial polytope obtained in this way starting with a d-dimensional simplex Let P be a d-dimensional staced polytope with n vertices Hibi and Terai [6] showed that β i,j ([ (P] = 0 unless i = j or i = j d + Since β i,i ([ (P] = β n i d+,n i ([ (P], it is sufficient to determine β i,i ([ (P] to find all β i,j ([ (P] In the same paper, they found the following formula for β, ([ (P]: ( n d β, ([ (P] = ( (3 Herzog and Li Marzi [4] gave another proof of (3 The main purpose of this paper is to prove (3 combinatorially In the meanwhile, we get as corollaries the results of Bruns and Hibi [2] : a formula of b ( if is a tree (or a cycle considered as a -dimensional simplicial complex 2 Definition of t-connected sum In this section we define a t-connected sum of simplicial complexes, which gives another equivalent definition of the boundary complex of a staced polytope See [3] for the details of connected sums And then, we extend the definition of t-connected sum to graphs, which has less restrictions on the construction Every graph in this paper is simple 2 A t-connected sum of simplicial complexes Let V and V be finite sets A relabeling is a bijection σ : V V If is a simplicial complex on V, then σ( = {σ(f : F } is a simplicial complex on V Definition 2 Let and 2 be simplicial complexes on V and V 2 respectively Let F and F 2 2 be maximal faces with F = F 2 Let V 2 be a finite set and σ : V 2 V 2 a relabeling such that V V 2 = F and σ(f 2 = F Then the connected sum # F,F 2 σ 2 of and 2 with respect to (F, F 2, σ is the simplicial complex ( σ( 2 \ {F } on V V 2 If = # F,F 2 σ 2 and F = F 2 = t, then we say that is a t-connected sum of and 2 Note that if and 2 are (d -dimensional pure simplicial complexes, ie, the dimension of each maximal face is d, then we can only define a d-connected sum of them Let, 2,, n be simplicial complexes A simplicial complex is said to be a t- connected sum of,, n if there is a sequence of simplicial complexes, 2,, n such that =, i is a t-connected sum of i and i for i = 2, 3,, n, and n = the electronic journal of combinatorics 7 (200, #R9 3

4 = = G( # 2 = G( #G( 2 = Figure : The -seleton of a 2-connected sum of and 2 is not a 2-connected sum of G( and G( 2 22 A t-connected sum of graphs Let G be a graph with vertex set V and edge set E Let W V Then the induced subgraph G W of G with respect to W is the graph with vertex set W and edge set {{x, y} E : x, y W } Let b (G = W V W = 3 (cc(g W, where cc(g W denotes the number of connected components of G W Let be a simplicial complex on V The -seleton G( of is the graph with vertex set V and edge set E = {F : F = 2} By definition, the connected components of W and G( W are identical for all W V Thus b ( = b (G( Now we define a t-connected sum of two graphs Definition 22 Let G and G 2 be graphs with vertex sets V and V 2, and edge sets E and E 2 respectively Let F V and F 2 V 2 be sets of vertices such that F = F 2, and G F and G 2 F2 are complete graphs Let V 2 be a finite set and σ : V 2 V 2 a relabeling such that V V 2 = F and σ(f 2 = F Then the connected sum G # F,F 2 σ G 2 of G and G 2 with respect to (F, F 2, σ is the graph with vertex set V V 2 and edge set E σ(e 2, where σ(e 2 = {{σ(x, σ(y} : {x, y} E 2 } If G = G # F,F 2 σ G 2 and F = F 2 = t, then we say that G is a t-connected sum of G and G 2 Note that in contrary to the definition of t-connected sum of simplicial complexes, it is not required that F and F 2 are maximal, and we do not remove any element in E σ(e 2 We define a t-connected sum of G, G 2,,G n as we did for simplicial complexes It is easy to see that, if F = F 2 3, then G( # F,F 2 σ 2 = G( # F,F 2 σ G( 2 Thus we get the following proposition Proposition 23 For t 3, if is a t-connected sum of, 2,, n, then G( is a t-connected sum of G(, G( 2,,G( n Note that Proposition 23 is not true if t = 2 as the following example shows Example 24 Let = {2, 23, 3} and 2 = {3, 34, 4} be simplicial complexes on V = {, 2, 3} and V 2 = {, 3, 4} Here 2 means the set {, 2} Let F = F 2 = {, 3} and let σ be the identity map from V 2 to itself Then the edge set of G( # F,F 2 σ 2 is {2, 23, 34, 4}, but the edge set of G( # F,F 2 σ G( 2 is {2, 23, 34, 4, 3} See Figure 4 4 the electronic journal of combinatorics 7 (200, #R9 4

5 3 Main results In this section we find a formula of b (G for a graph G which is a t-connected sum of two graphs To do this let us introduce the following notation For a graph G with vertex set V, let Note that c (G = b (G + ( V c (G = W V W = cc(g W Lemma 3 Let G and G 2 be graphs with n and n 2 vertices respectively Let t be a positive integer and let G be a t-connected sum of G and G 2 Then ( ( ( n2 t n t c (G = c i (G + c i (G 2 i i i=0 ( ( n + n 2 t n + n 2 2t + Proof Let V (resp V 2 be the vertex set of G (resp G 2 We have G = G # F,F 2 σ G 2 for some F V, F 2 V 2, a vertex set V 2 and a relabeling σ : V V 2 such that V V 2 = F, σ(f 2 = F, and G F and G 2 F2 are complete graphs on t vertices Let A be the set of pairs (C, W such that W V V 2, W = and C is a connected component of G W Let A = {(C, W A : C V }, A 2 = {(C, W A : C V 2 } Then c (G = A = A + A 2 A A 2 It is sufficient to show that A = i=0 c i(g ( n 2 i, A2 = i=0 c i(g 2 ( n i and A A 2 = ( n +n 2 ( n +n 2 2t Let B be the set of triples (C, W, X such that W V, X V 2 \V, X + W = and C is a connected component of G W Let φ : A B be the map defined by φ (C, W = (C V, W V, W \ V Then φ has the inverse map defined as follows For a triple (C, W, X B, φ (C, W, X = (C, W, where W = W X and C is the connected component of G W containing C Thus φ is a bijection and we get A = B = i=0 c i(g ( n 2 i Similarly we get A2 = i=0 c i(g 2 ( n i Now let B = {W V V 2 : W F } Let ψ : A A 2 B be the map defined by ψ(c, W = W We have the inverse map ψ as follows For W B, ψ (W = (C, W, where C is the connected component of G W containing W F, which is guaranteed to exist since G F = G F is a complete graph Thus ψ is a bijection, and we get A A 2 = B = ( n +n 2 ( n +n 2 2t Theorem 32 Let G and G 2 be graphs with n and n 2 vertices respectively Let t be a positive integer and let G be a t-connected sum of G and G 2 Then b (G = i=0 ( ( n2 t b i (G i ( ( n t n + n 2 2t + b i (G 2 + i the electronic journal of combinatorics 7 (200, #R9 5

6 Proof Since c (G = b (G+ ( n +n 2, ci (G = b i (G + ( n i and ci (G 2 = b i (G 2 + ( n 2 i, by Lemma 3, it is sufficient to show that ( n + n 2 t 2 = i=0 (( n i ( n2 t i + ( n2 which is immediate from the identity ( a ( b ( i=0 i i = a+b i ( n t, i Recall that a t-connected sum G of two graphs depends on the choice of vertices of each graph and the identification of the chosen vertices However, Theorem 32 says that b (G does not depend on them Thus we get the following important property of a t-connected sum of graphs Corollary 33 Let t be a positive integer and let G be a t-connected sum of graphs G, G 2,,G n If H is also a t-connected sum of G, G 2,,G n, then b (G = b (H for all Using Proposition 23, we get a formula for the special graded Betti number of a t-connected sum of two simplicial complexes for t 3 Corollary 34 Let and 2 be simplicial complexes on V and V 2 respectively with V = n and V 2 = n 2 Let t be a positive integer and let be a t-connected sum of and 2 If t 3, then b ( = ( ( ( ( n2 t n t n + n 2 2t b i ( + b i ( 2 + i i i=0 For an integer n, let K n denote a complete graph with n vertices Let G be a graph with vertex set V If H is a t-connected sum of G and K t+, then H is a graph obtained from G by adding a new vertex v connected to all vertices in W for some W V such that G W is isomorphic to K t Thus H is determined by choosing such a subset W V Using this observation, we get the following lemma Theorem 35 Let t be a positive integer Let G be a t-connected sum of n K t+ s Then ( n b (G = ( Proof We construct a sequence of graphs H,,H n as follows Let H be the complete graph with vertex set {v, v 2,,v t+ } For i 2, let H i be the graph obtained from H i by adding a new vertex v t+i connected to all vertices in {v, v 2,,v t } Then H n is a t-connected sum of n K t+ s, and we have b (G = b (H n by Corollary 33 In H n, the vertex v i is connected to all the other vertices for i t, and v j and v j are not connected to each other for all t + j, j t + n Thus b (H n = ( ( n Observe that every tree with n + vertices is a -connected sum of n K 2 s Thus we get the following nontrivial property of trees which was observed by Bruns and Hibi [2] the electronic journal of combinatorics 7 (200, #R9 6

7 Corollary 36 [2, Example 2 (b] Let T be a tree with n + vertices Then b (T does not depend on the specific tree T We have ( n b (T = ( Corollary 37 [2, Example 2 (c] Let G be an n-gon If = n, then b (G = 0; otherwise, ( n( n 2 b (G = n Proof It is clear for = n Assume < n Let V = {v,,v n } be the vertex set of G Then (n b (G = (cc(g W W V v V \W W = (cc(g W = v V = v V W V \{v} W = b (G V \{v} Since each G V \{v} is a tree with n vertices, we are done by Corollary 36 Remar 38 Bruns and Hibi [2] obtained Corollary 36 and Corollary 37 by showing that if is a tree (or an n-gon, considered as a -dimensional simplicial complex, then [ ] has a pure resolution Since [ ] is Cohen-Macaulay and it has a pure resolution, the Betti numbers are determined by its type (cf [] Now we can prove (3 Note that, for d 3, if P is a d-dimensional simplicial polytope and Q is a simplicial polytope obtained from P by attaching a d-dimensional simplex S to a facet of P, then (Q is a d-connected sum of (P and (S, and thus the - seleton G( (Q is a d-connected sum of G( (P and K d+ Hence the -seleton of the boundary complex of a d-dimensional staced polytope is a d-connected sum of K d+ s Theorem 39 Let P be a d-dimensional staced polytope with n vertices If d 3, then ( n d b ( (P = ( If d = 2, then { 0, if = n, b ( (P =, otherwise n( n Proof Assume d 3 Then the -seleton G( (P is a d-connected sum of n d K d+ s Thus by Theorem 35, we get b ( (P = b (G( (P = ( ( n d Now assume d = 2 Then G( (P is an n-gon Thus by Corollary 37 we are done ( n 2 the electronic journal of combinatorics 7 (200, #R9 7

8 References [] Winfried Bruns and Jürgen Herzog Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 993 [2] Winfried Bruns and Taayui Hibi Cohen-Macaulay partially ordered sets with pure resolutions European J Combin, 9(7: , 998 [3] Victor M Buchstaber and Taras E Panov Torus actions and their applications in topology and combinatorics, volume 24 of University Lecture Series American Mathematical Society, Providence, RI, 2002 [4] Jürgen Herzog and Enzo Maria Li Marzi Bounds for the Betti numbers of shellable simplicial complexes and polytopes In Commutative algebra and algebraic geometry (Ferrara, volume 206 of Lecture Notes in Pure and Appl Math, pages Deer, New Yor, 999 [5] Richard P Stanley Combinatorics and commutative algebra, volume 4 of Progress in Mathematics Birhäuser Boston Inc, Boston, MA, second edition, 996 [6] Naoi Terai and Taayui Hibi Computation of Betti numbers of monomial ideals associated with staced polytopes Manuscripta Math, 92(4: , 997 [7] Günter M Ziegler Lectures on polytopes, volume 52 of Graduate Texts in Mathematics Springer-Verlag, New Yor, 995 the electronic journal of combinatorics 7 (200, #R9 8