Remarks on Missing Faces and Generalized Lower Bounds on Face Numbers Eran Nevo∗ Department of Mathematics Cornell University, Ithaca, USA [email protected] Submitted: Oct 27, 2008; Accepted: Apr 18, 2009; Published: Apr 27, 2009 Mathematics Subject Classification: 52B11, 05A15. Dedicated to Anders Bj¨orner on the occasion of his 60th birthday. Abstract We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen’s generalized lower bounds, and of Barnette’s lower bounds, are conjectured for these families of complexes. Partial results on these conjectures are presented. 1 Introduction For simplicial polytopes, McMullen’s generalized lower bounds on face numbers were proved by Stanley, conveniently phrased as nonnegativity of the corresponding g-vector [16]. As the matrix that sends the g-vector to its f-vector has nonnegative entries, Stan- ley’s result immediately implies Barnette’s lower bound theorem for simplicial polytopes. In turn, Barnette’s result immediately implies that the simplex minimizes all face numbers among simplicial polytopes with the same dimension. We will phrase conjectures analogous to these three results, depending on a new pa- rameter, namely the maximal size of a missing face, i.e. the maximal size of a minimal non face with respect to inclusion. This gives a hierarchy of conjectures on lower bounds on face numbers, interpolating between the generalized lower bound conjecture for simplicial spheres [12] and Gal’s conjecture for flag spheres [8]. We will now work our way to these three conjectures, from weakest to strongest. It is well known, and easy to prove, that among all simplicial complexes with a nonzero (reduced) d-homology, the boundary of the (d + 1)-simplex minimizes all face numbers. Similarly, Meshulam proved that among all flag complexes with a nonzero (reduced) ∗Research partially supported by an NSF Award DMS-0757828. the electronic journal of combinatorics 16(2) (2009), #R8 1 d-homology, the boundary of the (d + 1)-dimensional crosspolytope minimizes all face numbers [13]. (Recently Athanasiadis [1] proved that for the subfamily of flag homology d-spheres the h-vector is minimized by the boundary of the (d+1)-dimensional crosspoly- tope, hence so are the face numbers.) For this minimization problem we can clearly assume that the complexes are d- dimensional, as restricting to the d-skeleton cannot make the d-th homology vanish and cannot increase the face numbers. We find it natural to view these two families of simplicial complexes as extreme cases of the following families. Let C(i, d) be the family of d-dimensional simplicial complexes with a nonzero reduced d-homology (if ∆ ∈ C(i, d) then H˜d(∆; Z) =6 0) and with no missing faces of dimension > i. (F is a missing face of ∆ if its boundary ∂F ⊆ ∆ and F∈ / ∆. Its dimension is |F |−1.) Thus, C(1,d) are the flag d-complexes with nonzero d-th homology and C(d +1,d) are all the d-complexes with a nonzero d-th homology. (Clearly if i>d + 1 then C(i, d)= C(d +1,d).) Denote by fi(∆) the number of i-dimensional faces in the complex ∆. Let d ≥ 0, 0 < i be integers. Then there exist unique integers q ≥ 0, 1 ≤ r ≤ i such that d +1= qi + r. (Note that the range 1 ≤ r ≤ i is unusual. It will simplify the writing later on.) Let S(i, d) := ∂σi ∗ ... ∗ ∂σi ∗ ∂σr, where ∂σi, the boundary of the i-simplex, appears q times in this join. Then S(i, d) is a d-dimensional simplicial sphere. Inspired by Meshulam [13], we prove that it has the following extremal properties: Theorem 1.1. Let d ≥ 0, 0 < i ≤ d +1 be integers. Write d +1= qi + r where 1 ≤ r ≤ i and q,r are integers. Let ∆ ∈ C(i, d). Then: (a) If i divides d +1 then fj(∆) ≥ fj(S(i, d)) for every j. (b) For any i, f0(∆) ≥ f0(S(i, d)). (c) For any i, fj(∆) ≥ fj(S(i, d)) for every 1 ≤ j ≤ r. (d) If i divides d +1 and fj(∆) = fj(S(i, d)) for every j then ∆= S(i, d). The cases i = d + 1 and i = 1 recover the two known results mentioned above. We write (b) and (c) separately on purpose, as (b) will play a special role. The condition i | (d +1) in part (a) seems to be an artifact of the proof. Conjecture 1.2. Let d ≥ 0 and 0 < i ≤ d +1 be integers. Let ∆ ∈ C(i, d). Then fj(∆) ≥ fj(S(i, d)) for every j. Moreover, if equality is attained for every j then ∆= S(i, d). A refined question is to give lower bounds on face numbers of complexes in C(i, d) with a given number of vertices. The answer to this question for simplicial polytopes is a well known result by Barnette, often referred to as ‘The lower bound theorem’ [3]. Barnette later showed that these lower bounds hold for all triangulated manifolds [2]. Kalai showed they hold for all homology spheres, and more general complexes, and characterized the case of equality [10]. the electronic journal of combinatorics 16(2) (2009), #R8 2 To state these results, we define stacked polytopes and homology spheres. A stacking is the operation of adding a pyramid over a facet of a given simplicial polytope. A polytope is stacked if it can be obtained from a simplex by repeating the stacking operation finitely many times. Let Sk(d, n) be the boundary complex of a stacked (d + 1)-polytope with n vertices. While the combinatorial type of Sk(d, n) is not unique, its face numbers are determined. Next, a d-dimensional complex ∆ is a homology sphere if for every face F in ∆ (including the empty set), and for every 0 ≤ j, there is an isomorphism of dim(∆)−|F | m reduced homology groups H˜j(lk(F, ∆); Z) ∼= H˜j(S ; Z) where S denotes the m- dimensional sphere, Z the integers and lk(F, ∆) is the link of F in ∆. In particular, the boundary complex of a simplicial polytope is a homology sphere; however there are many non-polytopal examples of homology spheres, e.g. [11]. The following is the lower bound theorem (LBT): Theorem 1.3. ([2, 3] and [10]) Let d ≥ 3, and let ∆ be the boundary complex of a simplicial (d +1)-polytope, or more generally a homology d-sphere, with n vertices. Then fj(∆) ≥ fj(Sk(d, n)) for every j. If equality holds for some j ≥ 1 then ∆ is combinatorially isomorphic to some Sk(d, n). We now seek an analogue of this result when an upper bound on the dimension of missing faces is specified. Let HS(i, d, n) be the family of d-dimensional homology spheres with n vertices and without missing faces of dimension > i. Let d ≥ 0, 0 < i and d +1 = qi + r as before (1 ≤ r ≤ i). If HS(i, d, n) =6 ∅, then by Theorem 1.1(b), n ≥ q(i+1)+(r+1). Hence the following definition makes sense: S(i, d, n) := ∂σi∗...∗∂σi ∗ Sk(r −1, n−q(i+1)), where ∂σi appears q times in this join. This is possible unless r =1 and n > q(i+1)+2. In the later case define S(i, d, n)= ∂σi∗...∗∂σi∗Sk(i, n−(q−1)(i+1)), where ∂σi appears q − 1 times in this join. In any case, S(i, d, n) ∈ HS(i, d, n). Conjecture 1.4. If ∆ ∈ HS(i, d, n) then fj(∆) ≥ fj(S(i, d, n)) for every j. Clearly Conjecture 1.4 implies Conjecture 1.2 restricted to homology spheres. For i = d + 1 and i = d the assertion of the conjecture is the LBT, Theorem 1.3. For i = d − 1 the conjecture holds if ∆ is the boundary of a simplicial polytope, and follows from the celebrated g-theorem [4, 16]. Here Stanley’s result is used. Surprisingly, in this case equalities for all j’s in the conjecture imply that ∆ = S(d − 1,d,n). This follows from recent results in [15]. For i = 1, the conjectured lower bounds (for flag homology spheres) would follow from Gal’s conjecture on the γ-polynomial [8, Conjecture 2.1.7]. In this case there are many examples of equalities for all j’s in the conjecture. We now relate the g-theorem and Gal’s conjecture, by defining new ‘g-vectors’ suitable for the families of d-dimensional homology spheres without missing faces of dimension > i, denoted by HS(i, d). Let d ≥ 0 and i > 0 be integers, and let q ≥ 0, 1 ≤ r ≤ i be the unique integers such that d +1= qi + r. For such d and i define the polynomial i q r Pd,i(t) := (1+ t + ... + t ) (1 + t + ... + t ). the electronic journal of combinatorics 16(2) (2009), #R8 3 It is symmetric as a multiplication of symmetric polynomials. Further, denote P−1,i := 1 (a constant polynomial). Define the ordered set of polynomials d 2 ⌊ +1 ⌋ Bd,i := (Pd,i(t), tPd−2,i(t), t Pd−4,i(t), ..., t 2 P d+1 (t)). d−2⌊ 2 ⌋,i Note that Bd,i is a basis for the space of symmetric polynomials of degree at most d+1 and d+1 axis of symmetry at ‘degree’ 2 (over the rationales, say). For a symmetric polynomial (d,i) (d,i) (d,i) h(t) in this space, let g (h(t)) = (g0 , ..., g d+1 ) be the vector of coefficients in the ⌊ 2 ⌋ expansion of h(t) in the basis Bd,i.