Problems on Simplicial Complexes and their Face Rings In the problems below, A always denotes a finite simplicial complex of dimension d — 1 with vertex set V and face ring fc[A]. Properties like "acyclic," "Cohen-Macaulay," etc., are always taken with respect to a fixed ground field k. 1. Let A be a simplicial complex with 159 3-dimensional faces. For i < 2, find the smallest possible number of i~faces of A. For i > 4, find the largest possible number of i-faces of A. 2. A {d — l)-dimensional simplicial complex A is pure if all its facets (= maximal faces) have dimension d — 1. Let (/o, /ij • • •, fd-i) be the /-vector of a pure {d — l)-dimensional simplicial complex. (a) Show that the vector (/d_2, /d-a,..., /o, 1) is the /-vector of a simplicial complex. (b) Show that fi < fd-2-i for -1 < i < [{d - 3)/2j, and that fo < fl < '" < f[id-l)/2i' (c) (unsolved) Characterize (or at least obtain significant new con­ ditions on) the /-vector of a pure simplicial complex. (A com­ plete characterization is probably hopeless, as it would give, for instance, all orders of finite projective planes.) 3. Give an example of a pure simplicial complex A which fails for some i to satisfy the "Upper Bound Inequality" hi < (^~^^*~M. Here n = /o(A), the number of vertices of A. (There exists such an example with /i(A) = (1,2,3,4,5, -26,23, -8,1).) 4. Give an example of two simplicial complexes Ai and A2 such that I All ^ IA2I, but such that /i(Ai) > 0 and /^(Aa) ^ 0. What is the smallest possible dimension of Ai and A2? (The symbol ^ denotes homeomorphism of topological spaces.) 5. Let A be a (d—l)-dimensional simplicial complex. For 0 < j < d—1, define the j-skeleton Aj of A by A^ = {F G A : dimF < j}. Express the /i-vector /i(Ad-2) in terms of /i(A). 6. Given integers n > d > 1, let A(n, d) be the simplicial complex with vertex set [n] whose facets consist of all d-subsets F of [n] with the 136 Problems on Simplicial Complexes the following property: If {i, i -h 1,..., j} C F, i — 1 0 F, j +1 ^ F, 2 > 1, and j < n, then j — i is odd. Find the /i-vector /i(A(n,d)) of A(n,rf). 7. Let r and A be simplicial complexes on disjoint vertex sets V and VK, respectively. Define the join F * A to be the simplicial complex on the vertex setVUW with faces FuG, where F G F and G € A. (If F consists of a single point, then F * A is the cone over A. If F consists of two disjoint points, then F * A is the suspension of A.) (a) Compute the /i-vector h{r * A) in terms of /i(F) and /i(A). (b) The boundary A(d) of the {d ~ I)-dimensional cross-polytope (as an abstract simplicial complex) has vertex set V = {xi,..., ^diVii" ",yd}^ with F C V a face of A(rf) if and only if {xuVi} 2 F for all i. Find h{A{d)). 8. Let A be a triangulation of a (d ~ l)-sphere, d> 3. (a) Show that the link of every vertex is simply-connected (i.e., the fundamental group of \\k{v)\ is trivial for every vertex v). (b) (very difficult) Show that for d > 6, the link of an edge need not be simply-connected. 9. Let A be a (d - l)-dimensional simplicial complex. (a) Let a : A —^ A be an automorphism of A of prime order p. Suppose that for all nonempty faces F of A we have a{F) ^ F. Let (/lo,..., /id) be the /i-vector of A. Show that /ii(A) = (-l)»(^^(modp). Deduce that x(^) = — 1 (mod p), (b) More generally, suppose G is a group of automorphisms of A, and let #G = g. Suppose that for all 1 ^^ a € G and 0 7?^ F G A we have (T(F) ^ F. Show that the congruences in (a) are still valid, with p replaced by g. 10. Let A be a (d — l)-dimensional simplicial complex with /-vector (/o? • • •) id-\) and /i-vector (fto? • • • ? ^d)* We say that A satisfies the Problems on Simplicial Complexes 137 Dehn'Sommerville equations if hi = hd-i for all i. Show that if d is even and A satisfies the Dehn-Sommerville equations, then /o - /i + /3 - /4 H- /e - A + /9 - /lo + • •' = 0. 11. Show that there does not exist a triangulation A of a ball B**""^ for d > 3 such that every facet intersects the boundary in a (rf — 2)-face together with the opposite vertex. 12. Let A be a triangulation of a (d— l)-dimensional ball, with h(A) = (/io,/ii,...,/irf)- (a) Show that h{dA) — {ho — hd, ho-\- hi — hd — hd-i, ho-i- hi -\- h2 — hd — hd-i — hd-2, ..., /lo -f h hd-i — hd — ' " — hi). (b) (unsolved) Show that hi > hd-i for 0 < i < [d/2j. (c) Show that hi = hd-i for 1 < z < d — 1 if and only if the boundary of A is the boundary of a simplex (i.e., has exactly d facets). (d) (unsolved) Characterize /i-vectors of balls. 13. Let A be a pure {d ~ l)-dimensional simplicial complex. A face of A of dimension < d — 1 is called free if it lies on exactly one facet. (a) If A is acyclic, must there exist a free face? What if A is contractible? (b) What if also A is shellable? 14. Show that if A is a shellable simplicial complex, then so is the barycentric subdivision sd(A). (We can define sd(A) to be the order complex of the poset of nonempty faces of A.) 15. Let A be a pure (d — l)-dimensional simplicial complex, with h- vector /i(A) = (/IQ, /ii,..., /i^). Let T be the order complex of the poset of nonempty faces of A. Thus F is a balanced complex with colors 1,2,..., d, where an i-dimensional face of A (regarded as a 138 Problems on Simplicial Complexes vertex of T) is colored i+1. Let 5 C [d]. Show that the flag /i-vector of r is given by t=0 where Dd^i{Sji + 1) denotes the number of permutations w = W1W2 ' • • Wd^i of [d+1] such that S := [rf+1] — 5 = {j : u/j > Wj^i] (the descent set of ti;) and K;d^_i = z + 1. 16. Find explicitly every simplicial complex A with the property that every ordering of its facets is a shelling. 17. Let P be a finite graded poset with ranks 1,2,...,n. Let S C [n]. Show that if P is shellable (i.e., the order complex 0{P) is shellable), then so is the rank-selected subposet P5. 18. A d'pyramid V is the convex hull of a (d— l)-polytope Q, called the basis of Vy and a point x ^ aff Q (= the affine span of Q), called the apex of P. Show that /i(P) = /i{Q) + /i-i(Q), with the convention /-i(Q) = /d-i(Q) = 1- 19. Let Q be a (d — l)-polytope and / a closed line segment such that relint(Q) n relint(/) is a single point. (Here relint denotes relative interior.) Then the d-polytope P = conv(Q U /) is called a d- bipyramid with basis Q, Show that /i(^) = /i(Q) + 2/i-i(Q), 0<z<d~2 /rf,l(P) = 2/,^2(Q). 20. Show that the Euler relation /o-/i+- • -f-(-l)''~7d~i = l+(-l)''"^ is the most general linear relation holding among the components of the /-vector of a d-polytope. Equivalently, the affine span in R*' of all /-vectors of rf-polytopes has codimension 1 (or dimension d~ 1). HINT: Use the previous two problems. 21. Show that the Dehn-Sommerville equations are the most general linear relations holding among the components of the /-vector (or /i-vector) of a simplicial d-polytope. Problems on Simplicial Complexes 139 22. A convex d-polytope V is cubical if every fax^et (or equivalently every proper face) is combinatorially equivalent to a cube of the appropriate dimension. Let Cd denote the set of all /-vectors of cubical d-polytopes. (a) Show that the afiine span aff Q of C has dimension at most [d/2\. (b) By constructing suitable examples, show that in fact aff Cd has dimension exactly [d/2j. (c) Show that a cubical polytope of dimension at least four has an even number of vertices. (The conclusion is false for three- dimensional cubical poly topes.) (d) (surprisingly difficult) Show that a cubical d-polytope has at least 2^ vertices. (e) Find a cubical 4-polytope for which /a > /o- (f) (unsolved) Given d and n, what is the maximum number of z-dimensional faces of a cubical cf-polytope with n vertices? 23. (generalizing (d) above) Suppose that the d-polytope V has no tri­ angular 2-face. Show that for every facet F of P, there is a facet G disjoint from F. Deduce that V has at least 2^ vertices. 24. Here we give two unsolved problems due to Kalai which illustrate our abysmal ignorance of certain aspects of convex poly topes. (a) Show that if P is a centrally-symmetric d-polytope, then /_i -h /o + • • • + /d-i > 3^. (b) Let j be a positive integer.