On the Homological Dimension of Lattices Roy Meshulam August, 008 Abstract Let L be a finite lattice and let L = L {ˆ0, ˆ1}. It is shown that if the order complex L satisfies H k L 0 then L k. Equality L = k holds iff L is isomorphic to the Boolean lattice {0, 1} k. 1 Introduction The order complex P of a finite poset P, is the simplicial complex on the vertex set P whose simplices are the increasing chains x 0 < < x k see [1] for a thorough discussion. A result of Kozlov [] implies that if the integral homology group H k P is non-trivial then P k + 1, with equality P = k + 1 iff P is isomorphic to the ordinal sum of k + 1 antichains of size. In this note we consider a similar question for lattices. For a lattice L with minimal element ˆ0 and maximal element ˆ1, let L = L {ˆ0, ˆ1}. Example: Let B k denote the Boolean lattice {0, 1} k. The order complex B k coincides with the barycentric subdivision of the boundary of the k 1-simplex, hence H k Bk 0. Here we show that B k is the unique smallest lattice with non-trivial k -dimensional homology. Theorem 1.1. If a lattice L satisfies H k L 0, then L k. Equality L = k holds iff L = B k. Department of Mathematics, Technion, Haifa 3000, Israel. e-mail: meshulam@math.technion.ac.il. Research supported by the Israel Science Foundation. 1
Let β i L denote the i-th reduced Betti number of L. Then µl, the Möbius function of L, is equal to the reduced Euler characteristic χ L = i 1i βi L. It is conjectured that if L = n then µl = on. Edelman and Kahn have shown see [5] that µl n logn. Theorem 1.1 implies the following slightly stronger result. Corollary 1.. If L is a lattice on n elements then β i L n log n. i In Section we establish Theorem 1.1 and Corollary 1. using an idea from the Edelman-Kahn proof of the upper bound on µl. We conclude the introduction with a conjecture concerning the relation between the reduced Betti numbers of L and the cardinality of L. For a prime power q, let L k q denote the lattice of linear subspaces of the k- dimensional vector space F k q over the field F q. Then β k Lk q = q k and Lk q = k [ k i=0 where [ k i]q i ]q = i 1 q k q j j=0 is the number of i- q i q j dimensional subspaces of F k q see e.g. [4]. Conjecture 1.3. If a lattice L satisfies β k L q k then L [ k i] k i=0 q. [ Note that lim k q 1 i ]q = k i. Hence Theorem 1.1 may be viewed as the q = 1 case of Conjecture 1.3. Homological Dimension of Lattices We prove Theorem 1.1 by induction on k. The case k = 1 is clear. Suppose k and let L be a lattice of minimal cardinality such that H k L 0. Let x be an arbitrary atom of L. The following observation is due to Quillen [3]. For completeness we include the proof. Claim.1. If x y for all coatoms y L, then L is contractible. Proof: Let P, Q be two posets. Recall see [1] that if f, g : P Q are two order-preserving maps such that fp gp for all p P, then the induced
simplicial maps f, g : P Q are homotopic. Now let P = Q = L and for p P let fp = x, gp = px and hp = p. Then fp gp hp for all p and thus Therefore L is contractible. constant map = f g h = identity map. By Claim.1 there exists a coatom y such that x y. Consider the lattices L = [ˆ0, y] and L = [x, ˆ1]. Then L L =, hence either L L or L L. By reversing the order if necessary, we may assume that L L. Let M be the lattice L {x}. Write A = L, B = [x, ˆ1, C = M and D = L. Then A = B C and B C = D. Note that H k B = 0 since B is a cone on x, and H k C = 0 by the minimality assumption on L. It thus follows from the Mayer-Vietoris sequence 0 = H k B H k C H k A H k 3 D that H k 3 D 0. Therefore by induction [x, ˆ1] = L k 1, hence L L k. Suppose now that L = k. The argument above implies that L = L = k 1 and that both H k 3 L and H k 3 L are non-trivial. Therefore by the induction hypothesis L = Bk 1 = L. Let x 1,..., x m denote the atoms of L, and for each 1 i m let y i be a coatom of L such that x i y i. Let L i = [ˆ0, y i ] and L i = [x i, ˆ1]. By the above it follows that L i L i = L and L i = B k 1 = L i. Note that the first statement implies that x i y j for i j. For a vector u = u 1,..., u k 1 {0, 1} k 1 = B k 1 and an ɛ {0, 1} let S ɛ u = {1 i k 1 : u i = ɛ}. Since x 1,..., x m 1 are the atoms of L m = B k 1, it follows that m = k and that the map φ 0 : B k 1 L k given by φ 0 u = i S 1 u x i is an isomorphism. Similarly, the map φ 1 : B k 1 L k given by φ 1 u = i S 0 u y i is an isomorphism. We now claim that the map φ : B k L given on u, ɛ {0, 1} k 1 {0, 1} by φu, ɛ = φ ɛ u is an isomorphism. φ is clearly bijective. It remains to show that φ is a lattice map. We will check that φ preserves the meet, i.e. φ ɛ ɛ u v = φ ɛ u φ ɛ v. 3
If ɛ = ɛ then this follows from the corresponding property of φ ɛ. For ɛ = 0, ɛ = 1 we have to show that x i = x i y j. 1 i S 1 u v i S 1 u j S 0 v The righthand side of 1 belongs to L m hence x i y j = i S 1 u j S 0 v i S 1 w for some w {0, 1} k 1. Since x i y i for all i it follows that w u v. The reverse inequality u v w is a consequence of the fact that x i y j for all i j. Therefore w = u v and thus 1 holds. The proof that φ preserves the join is similar. Proof of Corollary 1.: Let L = n. Since L is a complex on n vertices its Betti numbers satisfy β i L n 3 i+1 for all i. Theorem 1.1 on the other hand implies that β i L = 0 for i > log n. Hence β i L i i log n x i n 3 < n logn. i + 1 References [1] A. Björner, Topological methods, in Handbook of Combinatorics R. Graham, M. Grötschel, and L. Lovász, Eds., 1819 187, North- Holland, Amsterdam, 1995. [] D. Kozlov, Convex hulls of f- and β-vectors. Discrete Comput. Geom. 181997 41 431. [3] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. in Math. 81978 101 18. 4
[4] R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 1997. [5] G. Ziegler, Posets with maximal Möbius function, J. Combin. Theory Ser. A 561991 03. 5