Problems on Simplicial Complexes and Their Face Rings

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 Ddî{Sji + 1) denotes the number of permutations w = W1W2 ' • • Wdî of [d+1] such that S := [rf+1] — 5 = {j : u/j > Wjî] (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

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. Show that for d sufficiently large, every d-polytope has a j-face which is either a simplex or (is combinatorially equivalent to) a cube. (Kalai has shown that this statement is true for j = 2.)

25. Let A be an acyclic (over a field k, say) (d— l)-dimensional simpli cial complex. Let

F{x) = X: /.(A)a:'+^

(a) Show that F{x) is divisible by x -h 1. 140 Problems on Simplicial Complexes

(b) Let F{x) = (x 4- l)G(x). Show that G{x) has nonnegative coefficients. (HINT: Let • • • -^ Ci —• CO —• C_i ~* 0 be the augmented chain complex used to define the reduced homology groups Hi{A; k). What is the rank of the boundary operator

(c) Show that A can be partitioned into pairwise disjoint 2-element sets {F,F'} such that F C F' and \F' - F| = L (Note that this imphes (b).) (HINT: Let G be the Hasse diagram of the face poset P(A) of A, regarded as an undirected graph. G is a bipartite graph with vertex bipartition (VQ, V^i), where

K = {F€A : \F\ = i (mod 2)}.

We want to find a complete matching M in G. By the Marriage Theorem, M will exist if (and only if) every subset 5 of Vb is connected to at least |5| vertices in Vi. Assume S CVQ fails to satisfy this condition. Deduce that the fc-vector space kS with basis 5 contains a cycle (in the sense of reduced homology) which fails to be a boundary.) (d) Show that one can choose the partitioning into sets {F^F'} in (c) so that the faces F form a subcomplex of A. Deduce that (/o) • • 5 fd-i) is the /-vector of some acyclic simplicial complex A if and only if the polynomial G{x) = l+gox H h9(1-2^^"^ of (b) has the property that {go,..., gd-2) is the /-vector of some simplicial complex.

26. (a) Generalize the previous exercise as follows. A Betti set of a simplicial complex A is a subset J9 C A such that for all i,

{FE5:dimF = i}=A(A),

where ^t(A) = dimHi{A\ fc), the ith reduced Betti number of A (over fc). Show that A can be written as a disjoint union A = A' U J5 UQ, where (i) A' is a subcomplex of A, (ii) B is a Betti set, (iii) B is an antichain, i.e., if F, G € B and F Q G, then F = Gy (iv) A' U B is a subcomplex of A, and (v) there exists a bijection 7): A' —* Q such that for all F € A' we have F C v{F) and \r){F) - F| = L (b) Deduce a characterization of the /-vector of a simplicial com plex with given Betti numbers, generalizing Exercise 25(d). Problems on Simplicial Complexes 141

27. (a) A simplicial complex A is called j-acyclic if for every face F of A with \F\ < j, the link of F is acyclic. (Thus 0-acyclic just means that A is acyclic.) Let f(A) = (/o,/i,. • • ,/d-i)- Show that if A is j-acychc, then the polynomial D^{x) := 1 + Eo"^ /t(A)x*+i is divisible by (1 + xy+^ (b) Show that f = (/o? /ij • •»fd-\) is the /-vector of a j-acyclic simplicial complex if and only if there is a simplicial complex A' such that f = f (cr * A), where a is a j-dimensional simplex and * denotes join. Equivalently, we have

D^{x) = {i + xy+' f 1 + 'Y.\ix'A ,

where {go.gi,.,, ^gd-j-2) is the /-vector of some simplicial complex. (c) (unsolved) Can every j-acyclic simplicial complex A (regarded as a poset ordered by inclusion) be partitioned into disjoint intervals [F, F'] of length (or rank) j -h 1? (This would imply that DA{X)/{1 -f- xY'^^ is a polynomial with nonnegative coef ficients.) Moreover, can the intervals [F, F'] be chosen so that the set of their bottom elements F form a subcomplex of A? (This would imply (b).)

28. A convex polytope V is j-neighborly if every j vertices of V lie on a proper face of V. Let 7^ be a j-neighborly polytope with at least 7 + 1 vertices. Show that every j vertices of V are affinely independent. Deduce that V is z-neighborly for all 1 < i < j.

29. Let A be a j-neighborly Euler complex of dimension d — 1, with j > [d/2\. Show that A is the boundary of a simplex.

30. Show that every j-neighborly 2j-polytope is simplicial.

31. Suppose that A and A' are simplicial complexes whose face rings A:[A] and A:[A'] are isomorphic as graded algebras (or even as rings). Show that A and A' are isomorphic.

32. A graded algebra k[xi^..., Xn]/I is said to be a complete intersection if the ideal / is generated by a regular sequence. Find all simplicial complexes A for which the face ring k[A] is a complete intersection. 142 Problems on Simplicial Complexes

33. Give a direct proof that depth fc[A] = 1 if A is a disconnected simplicial complex.

34. Show that the depth of A:[A] is equal to the largest value of j for which the (j — l)-skeleton Aj_i is Cohen-Macaulay.

35. Show that if Ai and A2 are Cohen-Macaulay simplicial complexes, then so is their join Ai * A2.

36. (a) Given 1 < d < n, let An,d be the simphcial complex whose vertex set V consists of the squares of an n x d chessboard (so |V| = nd), and whose faces consist of subsets F of V with no two elements in the same row or column. Compute the /-vector and /i-vector of An,d- (b) Show that An,d is Cohen-Macaulay if and only if 2d < n -f 1. 37. Give an example of a (finite) simplicial complex A and a field k for which k[A] has no Ls.o.p.

38. Let A = [Gi, Fi] U • • • U [Gr, Fr] be a partitioning of the Cohen- Macaulay simplicial complex A. Let ^1,..., ^^ be an Ls.o.p. of k[A]. Show that the monomials x^* need not form a basis for fc[A] as a (free) module over fc[^i,... ,0d]. (There exists an example with d=2.)

39. Using Reisner's theorem, show that a Cohen-Macaulay complex is pure.

40. (a) Let P be a Cohen-Macaulay poset (i.e., the order complex A(P) is Cohen-Macaulay) of dimension d — 1, and let P = P U {6, i}. Let fjL denote the Mobius function of P. Write type(P) as short for type(A(P)). Suppose that for every x < y in P, either the order complex A(x,y) of the open interval {x,y) is doubly Cohen-Macaulay or else tJ>{x,y) = 0. Show that

type(P) = (-~l)''"^]^/i(a:o,a:i)/x(a:i,X2)---/i(x^_i,Xr),

where the sum is over all chains 6 = XQ < Xi < • • • < x^ = i in P such that /i(x,_i, x^+i) = 0 for all i, (b) Show that (finite) modular lattices satisfy the conditions of part (a). Problems on Simplicial Complexes 143

(c) Suppose P is a distributive (or even meet-distributive) lattice. A chain 0 = XQ < xi < • • • < x^ = 1 in P is called a Loewy chain if every interval [xt_i, Xj] is a boolean algebra. Show that type(P) is equal to the number of minimal Loewy chains in P (i.e., the number of chains C (containing 6 and i) which are Loewy chains, but for which any proper subset of C is not a Loewy chain). (d) Let t{a,b) = type(Ca4-i x Cfc+i), where Cm denotes an m- element chain. Let

P(x,y)= ^ t(a,6)xV. a,b>0

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2-Cohen-Macaulay, 71 category, 11 cd-index, 102 acyclic centrally-symmetric polytope, 119 simplicial complex, 22 chain complex, 13 space, 22 chamber, 105 additive, 16 Charney-Davis conjecture, 100 Alexander duality theorem, 66 cochain complex, 14 algebra with straightening law. 111 Cohen-Macaulay, 35 annihilator, 8 complex, 58 artinian, 10 poset, 100 ascending chain condition, 10 relative simplicial complex, 117 ASL, 111 space, 84 augmented coloring, 95 chain complex, 17 completely fundamental, 30 oriented chain complex, 20 core, 64 singular chain complex, 21 Courant function, 107 balanced simplicial complex, 95 Coxeter basis, 8 complex, 105 Betti number of a module, 46 group, 104 Birkhoff-von Neumann theorem, cross-polytope, 119 32 cyclic boolean poset, 112 module, 8 boundary complex, 75 polytope, 59 boundary degree-preserving homomorphism, of a manifold, 23 27 of a polytope, 6 Dehn-Sommerville equations, 59, of a pseudomanifold, 24 67, 115 Buchsbaum module, 73 depth, 35 building, 106 descending chain condition, 10 canonical descent, 119 h.s.o.p., 97 dimension of a polytope, 5 module, 49 domain, 7 162 Index

doubly Cohen-Macaulay, 71, 94 resolution, 14, 46 dual polytope, 6 functor, 11 dunce hat, 84 fundamental, 30 Eulerian poset, 101 /-vector, 54, 113 exact sequence, 9 p-conjecture, 76 exceptional set, 2 generically Gorenstein, 91 exceptional value, 3 geometric extendably shellable, 84 realization, 19 extending the scalars, 12 simplex, 128 extreme ray, 6 simplicial complex, 128 Ext, 15, 48 subdivision, 128 f.f.r., 46 Gorenstein, 50 face Gorenstein*, 67 of a cone, 6 Gorenstein sequence, 66 of a polytope, 5 graded of a simplicial complex, 19, 53 homomorphism, 27 of a singular simplex, 21 poset, 101 face Grobner basis, 110 monomial, 81 p-theorem, 76 poset, 101 h.s.o.p., 33 ring, 53, 113 hard Lefschetz theorem, 77 facet cover, 80 Hilbert facet of a polytope, 5 Basis Theorem, 6 fine grading, 54 function, 27 finite series, 27 element, 107 syzygy theorem, 46 free resolution, 46 Hodge algebra, 110 type, 9 homogeneous finitely-generated element, 26 algebra, 9 ideal, 27 module, 8 submodule, 27 fixed-point free, 121 system of parameters, 33 flag /-vector, 96 homological dimension, 46 flag /i-vector, 96 homology group, 14 flag complex, 100 Hom, 15 free /i-polynomial, 129 group action, 121 /i-vector, 55, 57, 58, 75, 113 involution, 121 module, 8 /-adic completion, 10 Index 163 injective monomial, 110 envelope, 16 M-sequence, 35 hull, 16, 48 Mobius function, 101 module, 15 multicomplex, 55 resolution, 16 M-vector, 56 integer stochastic matrix, 5, 25 negative support, 39 integral, 9 n-fold suspension, 24 integral polytope, 76 Noether Normalization Lemma, 9, interval, 79 34 irrelevant ideal, 26 noetherian, 6 join, 65 non-zero-divisor, 8 Koszul nonor lent able pseudomanifold, 24 complex, 18 order complex, 99 relation, 18 order ideal, 118 KruU dimension, 33 orientable Kruskal-Katona theorem, 55 manifold, 23 l.s.o.p., 77 pseudomanifold, 24 labelled poset, 118 oriented left-exact functor, 16 chain complex, 20 level ring, 91 simplex, 19 lexicographic shellability, 101, 119 parabolic subgroup, 105 linear extension, 119 partitionable, 80 link, 60 partitioning, 80 local partial h.s.o.p., 33 /i-polynomial, 129 piecewise polynomial, 107 /i-vector, 129 Pliicker relations, 112 face module, 131 Poincare Duality Theorem, 23 magic square, 5, 25 polyhedral manifold, 23 complex, 6 manifold with boundary, 23 subdivisions, 128 Matlis dual, 48 polytope, 5 matroid complex, 89 poset, 99 Mayer-Vietoris sequence, 22 of boolean type, 112 meet, 113 projective module, 13 meet-semilattice, 112 pseudomanifold, 24 minimal free resolution, 46 pure simplicial complex, 79 module, 7 g'-analogue, 106 of fractions, 13 quasi-geometric subdivision, 129 164 Index

quasiperiod, 4 chain complex, 21 quasipolynomial, 4 simplex, 21 radical, 7 skew Schubert variety, 112 rank of a module, 8 socle, 50 rank-selected subcomplex, 96 special l.s.o.p., 127, 131 rational polytope, 125 spline, 107 reciprocity of local cohomology, standard monomial, 110 42 Stanley-Reisner ring, 53 reduced star, 127 Euler characteristic, 20, 21 subdivision, 128 homology group, 17 supporting hyperplane, 5 support, 110 Mayer-Vietoris sequence, 22 suspension, 24 ring, 7 symmetry of a polytope, 125 singular cohomology group, 23 singular homology group, 21 tensor product, 12, 18 regular topological subdivision, 128 sequence, 35 toric variety, 76 subdivision, 133 Tor, 14 relative triangulation, 19, 128 homology, 22 type of a Cohen-Macaulay mod simplicial complex, 116 ule, 50, 90 restriction, 79, 81, 118, 129 right derived functor, 16 unimodal, 134 ring of fractions, 7 Upper Bound Conjecture, 59 vertex Schubert variety, 112 sequentially Cohen-Macaulay, 87 of a cone, 6 shellable, 79 of a polytope, 5 relative complex, 118 very free group action, 121 simplicial complex, 86 shelling, 79, 118 order, 79, 118 short exact sequence, 9 simplex, 19 simplicial complex, 19 polytope, 75 poset, 112 subdivision, 128 singular Printed in the United States 76245LV00001B/3 9 "780817 643690'