P.L.-Spheres, convex polytopes, and stress

Discrete & Computational Geometry, Apr 1996

We describe here the notion of generalized stress on simplicial complexes, which serves several purposes: it establishes a link between two proofs of the Lower Bound Theorem for simplicial convex polytopes; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.l.-spheres are Cohen-Macaulay; and is dual to the notion of McMullen's weights on simple polytopes which he used to give a simpler, more geometric proof of theg-theorem.

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P.L.-Spheres, convex polytopes, and stress

Discrete Comput Geom P.L.-Spheres 0 Convex Polytopes 0 Stress 0 0 Department of Mathematics,University of Kentucky , Lexington, KY 40506 , USA We describe here the notion of generalized stress on simplicial complexes, which serves several purposes: it establishes a link between two proofs oftbe Lower Bound Theorem for simplicial convex polytopes; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.1.-spheres are Cohen-Macaulay; and is dual to the notion of McMullen's weights on simple polytopes which he used to give a simpler, more geometric proof of the g-theorem. * Supported in part by NSF Grants DMS-8504050 and DMS-8802933, by NSA Grant MDA904-89-H2038, by the Mittag-LefflerInstitute,by DIMACS(Center for DiscreteMathematicsandTheoretical Computer Science),a National Science Foundation Science and Technology Center, NSF-STC88-09648,and by a grant from the EPSRC. - C. W. Lee 1. Introduction A b o u t 25 years ago two important extremal p r o b l e m s for convex polytopes were solved at almost the same time. M c M u l l e n [ 12 ] proved the U p p e r Bound Conjecture, which predicts the m a x i m u m n u m b e r of faces of each dimension that a convex d - p o l y t o p e (d-dimensional polytope) with n vertices can have. Barnette [ 1 ], [ 2 ] settled the L o w e r Bound Conjecture, which specifies the m i n i m u m number o f faces of each d i m e n s i o n that a simplicial convex d - p o l y t o p e with n vertices can possess. The first proofs o f these results were s o m e w h a t unrelated, but in the subsequent decade Stanley developed a c o m m o n algebraic perspective for recasting and ultimately reproving both of these results [ 21 ]. In fact, he established the complete characterization of face-vectors of simplicial (or dually, simple) polytopes (the g - t h e o r e m ) originally conjectured by McMullen. This quickly led to the further development of very powerful connections between the combinatorics of convex polytopes and the algebraic geometry of associated toric varieties [ 6 ], [ 16 ]. This interplay has proved to be very fruitful and far from exhausted, and many issues are as yet unresolved. Some important progress, as well as questions, center around the extensions of the face-counting results to other classes of objects such as nonsimplicial (or nonsimple) polytopes, unbounded polyhedra, or simplicial spheres. We describe here the notion of generalized stress, which serves several purposes: it establishes a link between two proofs of the Lower Bound Theorem; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.1.-spheres are Cohen-Macaulay; and is dual to the notion of McMullen's weights on simple polytopes which he used to give a simpler, more geometric proof of the g-theorem [ 13 ], [ 14 ]. Generalized stress was first introduced in [ 10 ], and a detailed overview was presented in [ 11 ]. 2. The L o w e r B o u n d T h e o r e m For a simplicial convex d-polytope P, let g2 = f , - d f o + ( d + l ) 2 " We begin by sketching two proofs of the Lower Bound Theorem, which states: T h e o r e m 1 (Barnette). For all simplicial polytopes, g2 is nonnegative. Here, fj- denotes the number of j-faces (j-dimensional faces) of P. The first proof is due to Stanley, the second to Kalai. 2.1. Stanley's Proof Stanley's [ 21 ] proof of this result is actually an easy corollary of his proof of the more powerful g-theorem, and requires some preliminary definitions. Let A be a simplicial (d - 1)-complex ((d - 1)-dimensional complex) on the vertex set {1. . . . . n }. The f -vector of A is the vector of nonnegative integers f = (f0 . . . . . fa-1), where 3~ denotes the number of faces (elements) of A of dimension j (cardinality j + 1). With the convention that f-1 = 1, the h-vector of A is the vector of integers h = (h0 . . . . . ha) defined by hk ~ (--1) j-k ( ; )J - k As is well known, the h-vector encodes the same amount o f information as the f-vector, since J3= J + ~ ( d - k d - j - 1 ) h k , j = - - I . . . . . d - - 1 . k=0 N o w define go = h0 = 1 and gk ----hk -- hk-1, k = 1. . . . . [ d / 2 J . The face ring of A over R is A = R[Xl . . . . . x n ] / l a , where It, is the ideal generated by all square-free monomials xi~ . . . xi, such that {il . . . . . is} is not a m e m b e r o f A. We grade A in a natural way by degree, A = A0 ~3 AI ~B A2 ~ 99-. For 01 . . . . . Od~ AI, define B = B0 ~ Bl fl) . . . . A/(01 . . . . . Od). Stanley [ 19 ], [ 20 ] proves: Theorem 2 (Stanley). A is Cohen-Macaulay if and only if Or. . . . . Od exist such that B = Bo ~ . . . ~ Bd and dim Bk = hk, k ~---0 . . . . . d. In this case the Oj can be chosen generically (i.e., with coefficients that are algebraically independent over R). I f the ring A is Cohen-Macaulay, then A is called a Cohen-Macaulay complex. Reisner [ 18 ] gives a homological characterization o f the class of C o h e n - M a c a u l a y complexes, which includes shellable simplicial complexes, simplicial balls and spheres, and boundary complexes o f simplicial polytopes. The h-vectors o f C o h e n - M a c a u l a y complexes are clearly nonnegative, but they must also satisfy certain nonlinear conditions. For01 . . . . . Od ~ A1, B = A / (01 . . . . . Od), and oo ~ B1, define C = Co ~BC1 ~3 . . . . B/(og). Stanley exploits a connection between the face ring o f a simplicial convex polytope and the cohomology o f an associated toric variety, and invokes the Hard Lefschetz Theorem for such varieties to prove: T h e o r e m 3 (Stanley). Suppose that A is the face ring of the boundary complex A of some simplicial convex d-polytope. Then O1. . . . . Od E A 1 and co ~ Bl exist such that: 1. B = B o ~ 3 . - . ~ Bd a n d d i m B k = hk, k = O . . . . . d. 2. Multiplication by coa-2k is a bijection between Bk and Ba-k, k = 0 . . . . . Ld/2J. In particular, multiplication by o9is an injection from Bk into Bk+t, k = 0 . . . . . Ld/2J - 1. As a consequence, C = Co fl) . . . ~ CLd/2j and gk = dim Ck, k = 0 . . . . . Ld/2J. A n immediate corollary is that the numbers gk are nonnegative, k = 0 . . . . . [ d / 2 ] . (This was first conjectured by McMullen and Walkup [ 15 ].) In particular, g2 > 0. We also see that h i -~ ha-i, i = 0 . . . . . [ d / 2 J . These are the Dehn-Sommerville relations, which can be proved directly by various combinatorial methods, and hold more generally for simplicial spheres. Stanley's theorem yields an explicit numerical characterization of the f - v e c t o r s o f simplicial d-polytopes, which is expressed in terms o f the hk and the gk (the gtheorem) [ 21 ]. T h e o r e m 4 (Stanley). Suppose that h = (h0 . . . . . hd)  Z d+l, go = h0, and gk = hk -- hk-1, k = 1. . . . . Ld/2J. Then h is the h-vector o f a simplicial d-polytope if and only if'. 1. hi = dd-i, i = 0 . . . . . ld/2J. c. w. Lee 2. gi > 0, i = 0 . . . . . Ld/2J. 3. go -- 1 and gi+l <- giIt? ' t9 ~--- 1. . . . . Ld/2J - 1. See, for example, [ 21 ] for the definition of the pseudopower gy). 2.2. Kalai's P r o o f Kalai's proof [ 8 ] that g2 is nonnegative is quite accessible, but does not have the full force of the g-theorem. Again, we need to start with some definitions. Let G = (V, E) be a graph, where V = {1. . . . . n}. Choose a point vi ~ R d for each vertex of the graph and make a bar-and-joint structure by placing bars connecting pairs of points corresponding to the edges of G (we do not worry about self-intersection). Often we refer to the vi themselves as the vertices, and the bars as the edges. A stress on this bar-and-joint structure is an assignment of numbers ~.ij to edges vi vj such that j: l)ivjEE )~ij(1)j -- Oi) = O (1) holds for every vertex /)i. The vector space of all stresses is the stress space of the structure. A n infinitesimal motion of the structure is a set of vectors U1. . . . . ~n e R d such that d([I (vi + t ~ i ) - (vj + t ~ j ) 112)/dt = 0 for all edges vi vj. Equivalently, (vi - vj) r (-6i- ~ j ) = 0 for all edges, or the projections of vi and ~j onto the affine span of {vi, vj } agree. Some infinitesimal motions are trivial in the sense that they are induced by rigid motions of R d itself. Motions apart from these are called nontrivial. If the structure admits only trivial motions, it is infinitesimally rigid. Using the classical relationship between the space of infinitesimal motions and the space of stresses of a structure, and the fact that the bar-and-joint structure associated with the edge-skeleton of a simplicial convex d-polytope P, d > 3, is infinitesimally rigid (where we take the vi to be the vertices of P itself), Kalai observes that the dimension of the stress space of P is g2, and hence g2 must be nonnegative. In this striking proof of the Lower Bound Theorem Kalai speculates whether it might be possible to extend the notions of stress and rigidity appropriately to the higherdimensional faces of P to reprove the nonnegativity of the other gk, and possibly even find a new proof of the g-theorem. The notion of generalized stress presented below accomplishes this, but these results depend in an essential way upon McMullen's new proof of the g-theorem [ 13 ], [ 14 ] using weights on simple polytopes. Generalized Stress 3.1. Working Toward a Definition We could define generalized stress by starting with some analog of classical stress or infinitesimal motion, but instead we work toward the definition by following the path by which 'it was originally discovered. This route was primarily motivated by attempts to mimic some aspects o f Kalai's algebraic shifting technique [ 7 ]. For x = (Xl . . . . . xn), and for (rl . . . . . rn) ~ Zn+, by x r we mean x p ...x~, ~ Define also s u p p x r = {i: ri ~ 0} (the support o f x r ) , r! = r l ! - - - r n ! , and Irl = rl + - - - + rn. Write ei for the vector o f length n consisting of all zeros, except for a one in the ith position, and e = (1 . . . . . 1). Let A be a simplicial complex (not necessarily of dimension d - 1) With n vertices {1. . . . . n}, and let R = R[xl . . . . . xn] = Ro (t) R1 (3 RZ (3 "'" be the ring o f polynomials, graded by degree. Consider any elements 01 . . . . . Od ~ R1. We wish to determine information about the dimension of Bk (as a vector space over R), where B = B o ( 3 B I ~ B 2 ~ . . . is the result of taking R and factoring out the ideal J = J o ~ J l ~ J 2 ( 3 . . " generated by IA and 01. . . . . Oa. Place an inner product on the vector space Rk by defining ( ~ r : Irl=k arxr' Z r : Irl=k brxr} : ~---.r:Irl=k arbr. W r i t e gk = Jk (]) J / . It is straightforward to see that )'-~: Irl=k brxr is in J ~ if and only if it is orthogonal to: 1. All monomials of the form xSx q where xq is square-free, suppxq r A, and I s l + l q l = k . 2. All polynomials of the form xsOj, where Isl = k - 1. Define vi = (oil . . . . . vial)r, i = 1. . . . . n, where Oj = ~-,i"=1 1)iJxi, J = 1. . . . . d. Then the first condition is equivalent to the condition b r = 0 if and the second condition is equivalent to the condition Define M to be the d x n matrix with columns ol . . . . . on. Then Y]r: IrL=kbrx~ satisfies condition (3) if and only if o r i=1 for every s ~ Z~_ such that Isl = k - 1. Thus we have a linear equation on the vectors 1)i for every such s. The second condition can be expressed more compactly if we look at where the left-hand side is to be regarded as a polynomial with vector coefficients, or n i=1 bs+e, vi = O, (2) (3) (4) This leads directly to our definition o f generalized linear stress: D e f i n i t i o n 1. Let A be a simplicial c o m p l e x (not necessarily o f d i m e n s i o n d - 1) on the set {1 . . . . . n}, and let Vl . . . . vn ~ R d. Let M be the d x n m a t r i x with columns Vl . . . . . on. F o r each k = 0, 1, 2 . . . . . a linear k-stress on A (with respect to Vl . . . . . Vn) is a p o l y n o m i a l o f the form that satisfies and b ( x ) = ~ r: Irl=k X r br'7. br = O i f The collection o f all linear k-stresses forms a vector space, which is d e n o t e d Ske.(In [ 10 ] we used the notation Bk.) There was evidence to suggest that the Hard Lefschetz element a~ in the p r o o f o f the g - t h e o r e m could be chosen to be Xl + - .- + xn. This was confirmed by M c M u l l e n [ 13 ], [ 14 ]. So we are also interested in the effect o f factoring out xl + - - - + x , from R as well. This suggests the definition of g e n e r a l i z e d affine stress: D e f i n i t i o n 2. Let A be a simplicial c o m p l e x (not necessarily o f d i m e n s i o n d - 1) on the set {1 . . . . . n}, and let Vl . . . . Vn E R d. Let M be the (d + 1) x n matrix obtained by appending a final row o f ones to the matrix M with columns vl . . . . . vn. For each k = 0, 1, 2 . . . . . an affine k-stress on A (with respect to vl . . . . . v.) is a p o l y n o m i a l o f the form o r The collection o f all affine k-stresses forms a vector space, which is d e n o t e d S~. (In [ 10 ] we used the notation Ck.) Equivalently, an affine k-stress is a linear k-stress that satisfies the additional condition o r for every s ~ Z~_ such that Isl = k - 1. That is, we have an affine relation on the vectors vi for every such s. It is obvious that b ( x ) is an affine k-stress with respect to vl . . . . . v~ if and only if it is a linear k-stress with respect to ~1 . . . . . Un, where ~ i=1 bs+e, = 0 , 1 Differential operators with constant coefficients acting on the stress spaces play an impoltant role. In particular, we can construct an operator that will provide a relationship between linear and affine stresses, and which is seen in Sections 9 and 10 to serve as the Lefschetz element in the p r o o f o f the g-theorem. For c ~ R n, define the function trr on the space of linear stresses by ar = c r v b = ~ i=1 n Ob ci 8xi for any linear stress b(x). Define in particular o) (b) ----(re (b) ---20b ~x/" T h e o r e m 6. Let A be any simplicial complex with n vertices, and let vl . . . . . Vn E R d. Then, f o r k = 1, 2, 3 . . . . . the function trc maps S~ into S~e_l, and, f o r k = 0, 1 , 2 . . . . . the kernel o f to restricted to S~ is S~. ~ a i i=O o x i OXo - / : l O x i a i P r o o f s ( f ) For and Conversely, suppose that/~(x0, xl . . . . . xn) ~ S~(A). For a polynomial expression f in x0 . . . . . xn, define s ( f ) = f(O, xl . . . . . xn). We can check that b = s(/0 is in S~(A): -- -- S = S = 0 . - - a i + ai a i s .~ Ob i=] OXi -~. v i ~ O . Why "Stress"? The use of the terms "linear" and "affine" in the definition has already been justified-the conditions for b(x) to be a stress involve either linear or affine relations on the vi. However, it is not yet clear why the term "stress" makes sense. This will be motivated in several stages. First we show that S~ is isomorphic to the classical stress space of a bar-and-joint structure. The higher dimensions will require some preliminary work. However, first, we consider some simple examples. 4.1. Examples The first example is an easy but important one that will resurface later in this paper. E x a m p l e 1. Consider a geometric d-simplex in R d and let A be its boundary complex. Choose Ol. . . . . Vd+l to be the vertices of the simplex itself. Assume further that the simplex is positioned such that no proper subset of the vertices is linearly dependent. Then nonzero ci e R exist such that d+l and all linear relations on the vi are nonzero scalar multiples o f this one. We claim that for all k = 0 . . . . . d, S~e is one-dimensional and is spanned by We can verify this by using the fact that E r: Irl=k X r c r y . r! d+l d+l E Cs+eil)i -~"cs E i=1 i=1 Cil)i = 0 for all s ~ Za++1 such that Is[ -----k - 1. Observe that cr is nonzero for all r. On the other hand, di"m S~e = 0 for all k > d, dim S~ = 1, and dim S~ = 0 for all k > 1, since the vi are affinely independent and so y ' f =+~ c i r E x a m p l e 2. Suppose that P is the standard octahedron in R 3 with vertices vl = (1, 0, 0) r, v: = ( - 1, 0, 0) r, v3 = (0, 1, 0) r, v4 = (0, - 1, 0) r, v5 = (0, 0, 1) r, and V6 ~ (0, 0, - 1) r. Then, for the boundary complex A o f P , the stress spaces with respect to Vl, . . . . v6 are given by: 1. S o e = R . 2. S~ is three-dimensional and has a basis {xl + xz, x3 + x4, x5 + x6}. 3. $2e is three-dimensional and has a basis {(xl + x2)(x3 + x4), (xl + x2)(x5 + x6), (X3 -'[- X4)(X 5 -'[- X6) }. . $3e is one-dimensional and has a basis {(xl + xe)(x3 + x4)(x5 + x6)}. 5. Se = {0} i f k > 3. 6. S~=R. 7. S~ is two-dimensional and has a basis {xl + x2 - x3 - x4, xl + x2 - x5 - x6}. 8. St = { 0 } i f k > 1. 4.2. Connection with Classical Stress Turning now to general simplicial complexes, we can describe the low-dimensional stress spaces and clarify the connection with classical stress: T h e o r e m 8. Let A be any simplicial complex with n vertices, and let vl . . . . . vn E R a. Then: 1. S ~ - - - - S ~ = R . 2. Sel is isomorphic to the space o f all linear relations on the vectors vl . . . . . vn. 3. S t is isomorphic to the space o f all affine relations on the vectors Vl . . . . . on. 4. S~ is isomorphic to the classical stress space on the bar-and-joint structure where the vertices are placed at the points vl . . . . . On, under the correspondence )~ij = bei + bej f o r all i ~ j . EL.-Spheres, Convex Polytopes, and Stress Proof. Only the fourth part requires any explanation. Assume b ~ S~. Set )~ij = be,+ej for all i, j = 1. . . . . n. Of c o u r s e , )~ij -~- )~ji, and )~ij = 0 if {i, j} is not an edge of A. From conditions (3) and (5) we find that, for all j = 1. . . . . n, i: i # j )~ij 1)i "at-)~jj 1)j i: i~j ~ . i j ( o i - vj), where E is the set of edges of A. Therefore the ~'ij s a t i s f y the equilibrium condition (1). Conversely, assume we have numbers ~.ij for each {i, j } ~ E that satisfy condition (1). For j = 1. . . . . n define and for i # j define bjj = ~ i: {i.j}~E )~ij, bei+eJ : {oi j ioftherwise.{i'j}eE' Reversing the previous calculations shows that the resulting quadratic polynomial b ( x ) is an affine 2-stress. [] 4.3. Coefficients o f Square-Free Terms Our next step is to show that under suitable conditions the coefficients of the square-free monomials of a linear or affine k-stress uniquely determine the remaining coefficients of the polynomial. We then concentrate our attention on the square-free terms, regarding the coefficients as assignments of numbers to various faces of the simplicial complex, and give a geometric necessary condition on these numbers that turns out to be a natural generalization of classical stress. We are, in fact, able to give explicit formulas for the coefficients of the non-square-free monomials in terms of the coefficients of the squarefree monomials, and in the process show that the above necessary condition is also sufficient and thus characterizes the coefficients of the square-free terms. For a simplicial complex A with n vertices and for Vl. . . . . v, ~ R d, we say that the vi are in linearly general position with respect to A if {vii . . . . . vi, } is linearly independent for every face {il . . . . . is} of A. T h e o r e m 9. Let A be any simplicial comptex with n vertices and assume that vl . . . . . v, are in linearly general position with respect to A. I f b(x) is a linear stress, then the coefficients o f the non-square-free monomials in b(x) are linear combinations o f the coefficients o f the square-free monomials and hence are uniquely determined by them. Proof. Let b ( x ) ~ S[. We use reverse induction on q = c a r d ( s u p p x r ) . The result is trivially true if q = k, so assume the result is true for some q such that 2 < q < k and suppose that c a r d ( s u p p x ~) = q - 1. Choose j such that rj > 1 and let s = r - ej. Condition (3) implies n E i=1 bs+e~vi = O. However, by the induction hypothesis the coefficients bs+e, are linear combinations o f the coefficients o f the square-free m o n o m i a l s when ri = 0, since c a r d ( s u p p x s+ei) = q in this case. This leaves the q - 1 coefficients bs+e, for i 6 supp x r to be uniquely d e t e r m i n e d since the corresponding vi are linearly independent by assumption. In particular, b~+ej = br is a linear combination o f the coefficients o f the square-free m o n o m i a l s . [] The above p r o o f shows how conditions (2) and (3) can be used in a systematic w a y to find all the coefficients o f b(x) if the coefficients o f the square-free terms are given. C o r o l l a r y 2. Let A be any simplicial complex with n vertices and let vl . . . . . vn E R a be chosen in a linearly general position with respect to A. Then dim S e =- 0 f o r all k > d i m A + l . P r o o f In the case that k > dim A + 1 there are no faces o f cardinality k, so all coefficients o f square-free m o n o m i a l s o f a linear k-stress must be zero. [] 4.4. A Geometrical Interpretation o f Stress For F = {ii . . . . . is} E A, define conv F (with respect to vl . . . . . vn) to be conv{vfi . . . . . or}. In an analogous way, define a f f F and span F . We sometimes abuse notation and write bF and x F for br and x r, respectively, where ri ----- 1 if i 6 F and ri = 0 i f / ~ F . We also use the notation F + i for F U {i} and F - i for F \ { i } . Finally, if i 6 F , by bF+i we mean br+ei, where r is as above. T h e o r e m 10. L e t A b e a n y s i m p l i c i a l ( d - 1 ) - c o m p l e x w i t h n v e r t i c e s a n d l e t v i . . . . . vn R a. Let b ( x ) be a linear (resp. affine) k-stress, k > 1. Choose any f a c e F o f A o f cardinality k - 1 and any point v in span F (resp. aft F). Then lies in span F (resp. aft F). Equivalently, if wi is the vector joining the projection o f vi onto span F (resp. aft F ) to vi, then bF+i(1)i -- 1)) i ~lk F bF+iWi -~- O. (6) Proof. Suppose that v E span F. Then, using condition (3), bF+i(1)i -- U) : 1)'-I- y ~ . b F + i V i - ~ i~lk F i~lk F bF+iV ~ 1 ) - - E b F + i l ~ i - iEF E iElk F bF+il) which is in span F. I f b is an affine stress, then by condition (5) the sum o f the coefficients in the above expression is E iElkF b F + i bF+i = 1. So we have an element o f aft F. Note that for linear k-stress, w i is the altitude vector for the point vi in the simplex conv({O} U ( F + i)), and for affine k-stress, wi is the altitude vector for the point vi in the simplex c o n v ( F + i). In particular, condition (6) for affine 2-stress is identical to condition (1) defining classical stress. So affine k-stress generalizes classical stress in a natural way, and could in fact have been defined by condition (6) in the first place. This is the definition that Kalai was thinking of (personal communication). Linear k-stress seems less natural at first sight since it is dependent upon choice o f origin. In the case o f simplicial polytopes, however, we see in Section 10 that linear stress becomes invariant under rigid motions when dualized and interpreted as McMullen's weights on simple polytopes. E x a m p l e 3. Let P be a simplicial convex d-polytope in R d, A its boundary complex, and Vl . . . . . Vn its vertices. Then the above theorem shows that dim S~ = 0. Take any b ( x ) E S~ and consider any subfacet F (i.e., o f cardinality d - 1). There are exactly two facets containing F and hence only two altitude vectors wi with respect to aff F , where i E lk F. By convexity these two vectors are not collinear and we know ~-~ bF+iWi ~ O, iElk F from which it follows that bF+i -~ 0 for i E l k F. Thus all the coefficients o f the squarefree monomials o f b ( x ) are zero, and so all o f the remaining coefficients must likewise be zero. 4.5. Formulas f o r the Coefficients Condition (6) is a nice geometrical necessary condition for the coefficients o f the squarefree terms of generalized stress. However (again with suitably general vi), this condition is also sufficient, as we now show. Assume A is a simplicial complex of dimension at most d - 1 with vertices 1. . . . . n, and that /)1 ..... On E R d are in linearly general position with respect to A, Assume further that Ul . . . . . Ud E R a. Suppose that G = {il . . . . . is} _c {1 . . . . . n}, where s < d. Fix an ordering o f the elements o f G and define Theorem 11. Let A be a simplicial complex on n vertices o f dimension at m o s t d - 1, let vl . . . . . vn be in linearly general position with respect to A , and let Ul . . . . . Ud be in linearly general position with respect to A and vl . . . . . on. Suppose that we have numbers be assigned to each (k - 1)-face F o f A that satisfy condition (6). For each r 9 Zn+ such that ]r[ = k and S = s u p p x r 9 A, define br = Y ~ (k - 1)-facesF containingS bF ~ I [ F - i]ri-1.. i~F Then b ( x ) = E r : ]rl=k b r ( x r / r .!) is a linear k-stress. Proof. We already know that there can be at most one linear k-stress b ( x ) with the given coefficients o f the square-free terms. We must show that in fact there is one, and that it is given by the formula above. Consider one instance o f condition (3): n y ~ bs+ejVj = O, j = l where [s[ = k - 1. Let S = s u p p x s. The coefficients bs+ej appearing in the expression correspond to monomials with support size either c a r d ( s u p p x s) (if j E S) or card(suppx s) + 1 (if j 9 lk S). So we can contemplate the possibility o f using these conditions repeatedly to determine the coefficients o f monomials with smaller supports from the coefficients o f monomials with larger supports. In the process we need to verify that: (i) For a given instance of the condition it is possible to solve for the unknown coefficients, i.e., that j elk S bs+ejVj 9 span{vj: j 9 S}. (7) (ii) I f the same coefficient is determined from two different applications o f condition (3) in this manner, that we do not get contradictory values. The p r o o f will therefore be by reverse induction on p = card(suppxr). The formula stated in the theorem is trivially true if k = 1 or if p = k so we assume that k > 2 and 1 < p < k. Choose any m for which rm > 1. Let s ----r - e m and S = supp x s = supp x r. EL:Spheres,ConvexPolytopes,and Stress tl E i=1 bs(j)+eil3i = O. Wedge this with Vs-q+j and sum over all j r S: Z jr ~ i=1 (bs(j)+eiVi A 1)s_q+j)= O. If i = j or if i ~ S - q, then vi A Vs_q+j ~- O. If i r S, then one of the terms in the above expression is However, interchanging the roles of i and j also yields the term These terms cancel since s ( j ) + ei = s(i) + ej. Looking at the remaining terms (where i = q) we see ~"~(bs(j)+eq 1)q A 1)S_q+j) = O. yes However, s ( j ) + eq = r - eq + ej + eq = r + ej, so bs(j)+e~ vi A vj A VS-q. bs(i)+ej vj A 73i A 1)S_q . 0 = ~_,(br+e, vj A VS) jCS Therefore Cramer's rule can be used to solve for br (which equals bs+ej when j = m) in the system Z j~s bs+e, V j = - - E b s + e , Vj, j~lkS d E b s + e j l 3 j 't- Z jES j=p+l c j u j = giving br IS - m + (-Y~7~tt s bs+ejvj ) ] is] Applying induction and using some Grassman-Plticker relations, The fact that this final formula is independent o f the choice o f m shows that we will not get contradictory values for br from different choices o f s. [] Filliman [ 4 ] and Tay et al. [ 22 ] have also shown the sufficiency o f condition (6), but without the explicit formula above. (Unfortunately, however, the proof o f the g - t h e o r e m for p.1.-spheres in [ 4 ] is incorrect.) The relationship between the space of classical stresses (affine 2-stress) and the space o f classical infinitesimal motions o f bar-and-joint structures is straightforward: they are the left and right nullspaces of a c o m m o n matrix. This suggests a natural way to define generalized infinitesimal motions associated with affine k-stress. Suppose that b(x) is an affine k-stress o f a simplicial complex A with respect to vl . . . . . Vn ~ R d. Looking at the equivalent condition on coefficients o f square-free terms, we have a number bF assigned to each (k - 1)-face F , such that, for every (k - 2)-face G, ~-~ bG+iWi ~- O, i~lkG EL:Spheres, Convex Polytopes, and Stress where wi is the altitude vector o f ~)i in the simplex conv(G + i). We can express these conditions in terms o f a matrix R whose rows are indexed by the (k - 1)-faces F and whose columns occur in groups of d with one group for each (k - 2)-face G. In the row corresponding to F and the group o f columns corresponding to G we place the row vector o f length d: O r w (rG,F) if if G ~ F, G C F. Here, W(G,V) is the altitude vector o f the simplex cony F with respect to conv G. Then the left nullspace o f R is S~. A n infinitesimal k-motion will then be defined to be an element o f the right nullspace o f R, and is described by an assignment of a vector ~ c E R d to each (k - 2)-face G such that, for every (k - 1)-face F , We have condensed the notation, writing wi for W~F-i,F) and -6i for v F - i . We can reformulate this condition by writing Gi for F -- i and ui for the unit outer normal vector o f conv Gi with respect to conv F in aff F. This yields ~ iEF w i .-vi = 0 . ~__j Ui Uwi II 9~i = 0, iEF 2 _ , ui 9 v o l k - 2 ( a i ) -- O. iEF ,7 .VOlk-2(Gi)(Ui "-mi) = O, iEF which upon dividing by volk_l ( F ) implies Therefore where mi = ~i/vol2_2(Gi). So an infinitesimal k-motion is a choice of vector ~ G E R d for each (k - 2)-face G such that GcF vOlk-2(G)(uG 9~ ) for every (k - 1)-face F , where u 6 is unit outer normal of conv G with respect to conv F in a f f F . T h e o r e m 12. Suppose that vectors ~ G are given f o r each (k - 2)-face G. Then the following two conditions are equivalent: 1. The ~ G constitute an infinitesimal k-motion. 2. For each (k - 1)-face F a vector mF E R a parallel to F exists such that'mF "UG = mG " U6 f o r all G C F. One direction is easy. Suppose that (2) holds. Then, for each (k - 1)-face F, E GcF volk-2(G)(UG" raG) = E GcF VOlk-2(G)(UG-mF) = O . m F by Minkowski's theorem. So (1) holds. On the other hand, suppose that (1) holds. Fix F and regard the UG and the m 6 as sitting naturally in R k-1 and constructthe (k - 1) x k matrix U whose columns are the u6. The rows of U are linearly independent, so the left nullspace of U has dimension one. Now the vector y whose entries are the volk_2(G) is in the nullspace of U, and the vector z whose entries are the UG 9~ G is orthogonal to y. Therefore z is in the rowspace of U, and in particular a single vector ~-F exists such that ~ e 9U = z. [] This gives us equivalent formulations of condition (8). Suppose that vectors ~ are given for each (k - 2)-face G. For a (k - 1)-face F and G C F, let m e be the projection o f m G onto the (k - 1)-dimensional linear space V parallel to F. For a real number t let F ( t ) be the (k - 1)-simplex determined by translating a f f G by the vector t m a . Corollary 3. The following conditions are each equivalent to condition (8): 1. Y~GcF volk-2(G)(u6 9m a ) = O. 2. F ( t ) is congruent to F. 3. F(1) is congruent to F. 4. ( d / d t ) vol2_l ( F ( t ) ) = O. Proof. (1) is clear since uG 9m a = u6 9m e , (2) and (3) hold since by the theorem we are equivalently translating each a f f G by the same vector mF. (4) then follows immediately. [] The last condition is a very natural generalization of the definition of classical infinitesimal motion (infinitesimal 2-motion) and was also observed by Filliman [ 4 ]. See [ 22 ] for a deeper study of the relationship between generalized stress and skeletal rigidity of cell complexes (not necessarily simplicial). 6. Bistellar Operations In this section we examine how the various stress spaces change under the action of certain local changes in a simplicial (d - 1)-complex A. Let F and G be disjoint nonempty subsets of {1. . . . . n} of cardinality p and q, respectively, such that p + q = d + 1, F E A, G ~ A, and lk F = OG = {G': G ' is a proper subset of G}, the boundary of G. EL-Spheres, ConvexPolytopes,and Stress The simplicial complex A' = ( A k F ) tO (G 9 OF) is the result of performing a bistellar operation on A. During this operation we remove all faces containing F and introduce all sets of the form G to F ' where F ' E OF. The faces of A' that are new are those faces of A' that contain G, and the faces of A which are lost are the faces of A that contain F. The local change in the structure of A induces a corresponding simple "local" change in the linear stress spaces. T h e o r e m 13. Assume A and A ' are as above and that vl . . . . . vn are in linearly general position with respect to both A and A'. Then Proof. The main idea is to use an intermediate simplicial complex A" to mediate the changes in the stress spaces. Define A" = A tO ( G . OF) and observe that A" also equals A' tO ( F . OG). We will show that [ d i m Sse(A) dim S*e(A") = / c t i m s [ ( z x ) + 1 if O < s < q - 1 , if q < s < d , and by symmetry ] dim Sst (A') dim S~e(A") = / d i m S~e(A') + 1 if 0 < s < p - 1 , if p < s < _ d . Since A and A" share the same faces of cardinality s when 0 _< s _< q - 1, then Sse(A) must be the same as Ses(A ") for these values of s. Assume that q _< s < d and take S = F D G, a subset of cardinality d + 1. All of the proper faces of S are in A". Define c ( x ) to he the unique (up to scalar multiple) linear s-stress on the simplicial complex consisting of all subsets of S as constructed in Example 1. Each face of A is also a face of A", so S,e(A) _ SaC(A"). Suppose that b ( x ) is a linear s-stress that is in A" but not in A. This implies that br is nonzero for some r such that suppx r E openstar G (the set of all faces of A" that contain G). We claim that the restriction of b to the faces of S must be a nonzero multiple of c(x); i.e., that there is a nonzero real number t such that br = tCr for all x r supported on openstar G, and hence b ( x ) - t c ( x ) is in Sse(A). Since G is the only face of cardinality q in openstar G, this is clearly true if s = q. So assume q + 1 < s < d. Choose any r such that br is nonzero and suppx r E openstar G. Since s > q, there is a j such that supp x r-ej ~ openstar G. Condition (3) implies that E i n= I b r-ej+ei 1)i -~" O. AS this sum involves only the d + 1 vectors vi such that i E S, the coefficients in this sum must be a common multiple of the corresponding coefficients of c(x). Using this procedure repeatedly to determine the other coefficients br verifies the claim. The resulting direct sum decomposition of Sse (A") establishes the change in dimension. [] 7.1. P.L.-Spheres 7. Simplicial Spheres, EL-Spheres, and Pseudomanifolds Bistellar operations are ideally suited for proofs by induction on p.1.-manifolds, especially considering Pachner's result [ 17 ] that any two p.1.-manifolds that are p.l.-homeomorphic can be transformed into each other by a sequence of bistellar operations. In particular, every p.1.-sphere can be obtained from the boundary of a simplex by such operations, and the discussion in the previous section almost immediately proves: Corollary 4. I f A is a simplicial p.l.-sphere, then A is Cohen-Macaulay. Proof Assume that A is a (d - 1)-dimensional simplicial p.l.-sphere with vertices {1. . . . . n}. Choose vl . . . . . vn E R d in linearly general position with respect to all subsets of {1. . . . . n} of cardinality d. The boundary of a (d - 1)-simplex is C o h e n Macaulay by part (1) of Corollary 1 and Example 1, since its h-vector equals (1 . . . . . 1). It is well known that the components of the h-vector of a simplicial complex change under the action of a bistellar operation in exactly the same way as the changes in the dimensions of the linear stress spaces described in Theorem 13. So since A can be obtained from the boundary of a simplex by a sequence of bistellar operations, we have hs(A) = dim S[ for all s. Therefore A is Cohen-Macaulay by Corollary 1. [] 7.2. Pseudomanifolds We now turn to a larger class of simplicial complexes which includes simplicial manifolds. A simplicial (d - 1)-complex is said to be a pseudomanifold if: (i) Every maximal face has dimension d - 1. (ii) Every (d - 2)-face is contained in exactly two faces of dimension d - 1. (iii) Any two (d - 1)-faces can be connected by a path of (d - 1)-faces, each two succeeding faces of which are adjacent (share a common (d - 2)-face). T h e o r e m 14. I f A is an orientable ( d - 1)-pseudomanifold on n vertices and vl . . . . . vn are in a linearly general position with respect to A, then dim S~(A) = 1. Actually, Tay et al. [ 23 ] prove the stronger result that the dimension of Sde(A) equals the dimension of the homology Hd ( A , R), and the proof of the above theorem hints why this is true. Proof Let b(x) be a linear d-stress on A. By Theorem 9 it suffices to study the squarefree coefficients of b(x). Choose a consistent orientation of all the facets ((d - 1)-faces) of A and use this to induce an ordering of the elements of each facet. Let G be a subfacet ((d - 2)-face) of A, and let F1 and F2 be the two facets containing G. Theorem 10 implies that [F1]bEF,~ = [Fz]bEF+ so a constant t exists such that bF  t[F] -1 for every EL.-Spheres, ConvexPolytopes, and Stress facet F. The coefficients of the non-square-free terms are then uniquely determined. So, up to scalar multiple, there is only one element in Se(A). [] In Section 9 the geometrical significance of this canonical linear d-stress (the one for which be = [F] -1 for each facet), in the case that A is the boundary complex of a simplicial convex polytope, will b e c o m e apparent. Suppose that A is a simplicial (d - 1)-complex on {1 . . . . . n} and Vl . . . . . vn ~ R a. For G = {il . . . . . is }, a subset of {1. . . . . n}, define the function r e on the space of linear stresses by In particular, write z t ( b ) = aSb Oxi~ . . . Oxi, " vi (b) . . . . Ob Oxi . T h e o r e m 15. Let A be a simplicial orientable (d - 1)-pseudomanifold on {1 . . . . . n} and let vl . . . . . vn E R d be in linearly general position with respect to A . Suppose that b ( x ) is the canonical linear d-stress o f A and G is a f a c e o f A o f cardinality s. Then rG(b) is a linear (d - s)-stress supported on clstar G. In the special case that l k G is a (d - s - 1)-sphere, then up to scalar multiple rG(b) is the unique nonzero linear (d - s)-stress supported on clstar G. Proof. The first part of the theorem is obvious: if the coefficient of x r is nonzero in rG (b), then the coefficient of Xil " . . x i , x r must be nonzero in b (x). Hence G U (supp x r) is a face of A and so s u p p x r ~ clstar G. For the second part, note that clstar G can be obtained by starting with Ik G and successively joining it to the vertices of G. Since lk G is a (d - s - D-sphere, it has a unique linear ( d - s)-stress (up to scalar multiple). By repeated application of T h e o r e m 7, so does clstar G. Now it is easy to see that z~(b) is nonzero since b ( x ) is, and so z t ( b ) must be a generator of the linear (d - s)-stresses on clstar G. [] 8. S h d l i n g s Consider a simplicial (d - 1)-complex on {1, . . . , n} such that every maximal face has dimension d - 1 (is a facet). Then the complex is said to be shellable if the facets can be ordered F1 . . . . . Fm in such a way that, for k = 1. . . . . m, there is a unique minimal face Gk that is in F~ but is not in Uik__-~~ i . Here, F i denotes the simplicial complex consisting of all subsets of F,-. It is well known that, as each facet Fk is added, precisely one component hs of the h-vector increases by one, the remaining components being unchanged; specifically, s = card Gk. Using this and understanding the changes in the face ring during the shelling, Kind and Kteinschmidt [ 9 ] give an inductive proof that shellable simplicial complexes are Cohen-Macaulay. It is also possible to use generalized stress to prove this result by showing that the dimension of Sse increases by one when Fk is added, while the dimensions of the other linear stress spaces do not change. However, we content ourselves with considering the special case when A is a simplicial (d - 1)-sphere. Assume that vl . . . . . Vn ~ R d are in linearly general position with respect to A, and let b ( x ) be the canonical linear d-stress on A. When Fk is added, the closed star of G~ = F k \ G k is completed. If card Gk = s, then card G~ = d - s, and Theorem 15 implies that up to scalar multiple there is a unique linear s-stress rG; (b) supported on clstar G~. The coefficient of this stress associated with the face Gk is nonzero, so this stress was not present before Fk was added. So we can use the shelling of A to derive a basis for the stress spaces. T h e o r e m 16. l f A is a sheUable simplicial (d - - 1)-sphere whose n vertices, vl . . . . . vn 9 R d, are in linearly generalposition with respect to A , and F1 . . . . . Fro, G] . . . . . G~ and b(x ) are as above, then {rr, (b): card G'k = d - s} is a basis f o r Ses. Hence the collection {rG(b): G is a f a c e o f A o f cardinality d - s} spans Se~. 9. Simplicial Convex Polytopes In this section we specialize further and consider the case when A is the boundary complex of some simplicial convex d-polytope P C R a. This was the motivating case for defining generalized stress in the first place and trying to understand the g-theorem. 9.1. Canonical Stress and Volume Assume that P contains the origin in its interior. Then the vertices vl . . . . . on of P are in linearly general position with respect to A. Since A is shellable, we know dim Se = hi, i = 0 . . . . . d. The definition of affine stress seems more geometrically natural for simplicial complexes since affine stress is invariant under translation. The linear stress spaces, while also geometrically definable, depend upon the choice of origin and change with translation. It turns out, however, that this situation changes entirely when we turn to the polar P* of P and describe the linear stresses in terms of conditions on P*. This will become clearer in Section 10, but already in this section we begin to see the significance of using the polar to understand stress. F o r x 9 Rn, consider the polytope Q ( x ) = {y 9 Rd: y T v i < Xi, i = 1. . . . . n}. O f course, Q(e) is the polar P* of P. Since P* is simple, for values o f x i n e a r 1, Q ( x ) and P* are strongly isomorphic. It is well known that the volume of Q ( x ) as a function of the xi is a homogeneous polynomial V ( x ) = )--~-r:Irl=db r ( x r / r ! ) of degree d, br = 0 whenever suppx r r P, and bF -~- [ F ] -1 for every facet F of P. The canonical linear d-stress b(x) on A also shares these properties, and so perhaps the following result is not completely unexpected: Theorem 17. Let P be as above. Then the canonical linear d-stress is precisely V (x ). F o r e v e r y u 9 R d, Q ( x I . . . . . Xn) -t- u = Q ( x 1 --I-uT1)I . . . . . Xn + uTvn) ( w e a r e O = [ V ( X l . . . . . Xn) -- V ( x 1 + uTo1 . . . . . X n + uTpn)] ~-~ br+eiUT tli i=1 u T \ i = l br+e Vi 9 ' / However, this is true for every u, so Y~-7=1br+e, Pi = 0 and V(x) is a linear d-stress. That V(x) is the same as the canonical linear d-stress follows from the fact that the coefficients of the square-free terms of V ( x ) agree with those o f the canonical linear d-stress. [] 9.2. Lower-Dimensional Canonical Stresses The above proof mimics the proof o f Minkowski's theorem that .n~vob_~(~)/)~i = o , which we already used in Section 5. (F1 . . . . . Fn are the facets o f P* corresponding to the vertices Vl. . . . . vn of P.) In fact, the relationship between Minkowski's theorem and stress is quite strong, as we will see. One way to prove the g-theorem would be to show that the application o f 09d-2i induces a bijection between Sed_iand S/e, i = 0 . . . . . [ d / 2 J . Actually, it would suffice to show that 09: S/e --+ Se_l is surjective for i = 1. . . . . Ld/2J. McMullen's new proof o f the g-theorem shows that the bijections proposed here are valid. Given the canonical linear d-stress V (x), we might consider applying 09 repeatedly to get canonical linear i-stresses 09d-i(V(x)), i -----0 . . . . . d -- 1. Let W ( x ) = V(xl + 1. . . . . xn + 1). Then, for small x, W ( x ) is the volume o f a polytope near P*. Write W(x) = )-]/d=l Wi(x), where Wi(x) is a homogeneous polynomial o f degree i, i = 0 . . . . . d. It is clear that the constant Wo(x) is the volume o f P* and Wd(X) = V(x). It is also easy to see that W1(x) = )--]~=1(VOld-1 (Fi)/ll vi II)xi. T h e o r e m 18. Let P be as above. Then J - i ( V (x)) = (d -- i)! Wi(x), i = 0 . . . . . d. Proof. We calculate the contribution o f br(xr/r!) in V ( x ) to the coefficient o f x s in Wi (x), where x s Ixr. Expanding br (Xl + 1) r~ " ' " (xn + 1) r" r l ! . . 9rn! we see that the contribution is On the other hand, the contribution of in V ( x ) to the coefficient o f x s in w d - i ( V ( x ) ) , where i = d -- [s[, is [] Corollary 5. Let P be as above. 1. The canonical linear O-stress ofl (V (x )) equals d! vol(P*). 2. The canonical linear 1-stress w d-1 (V (x )) equals (d - 1)! E i=l ?1 VOld-l(Fi) [11)i[I Xi" That is, the canonical linear combination of the vi induced by co is (up to scalar multiple) the same as that induced by Minkowski' s theorem. We can find the coefficient of the square-free term of Wi corresponding to an (i - 1)face F of P by looking at the corresponding (d - / ) - f a c e F* of P* and computing the contribution to the local change in the volume of P* due to the translations of the facets containing F*. This change depends upon the (d - / ) - v o l u m e of F* and the size of the cone of the associated normal vectors, rescaled to account for the fact that they may not be of unit length. Theorem 19. The coefficient of the square-free term of Wi corresponding to F is VOld_/ (F*) voli(conv({O} kJ {vi: i E F}))" See also [ 4 ]. So each Wi is associated in a very natural way with the (d - / ) - v o l u m e s of the (d - / ) - f a c e s of P*. Notice that we can write d W(x) = ~ i=0 Wi(x) = Z i=O = Z i=0 d t o d - i ( v ( x ) ) (d---i)--T d w i ( V ( x ) ) i! P.L.-Spheres,Convex Polytopes, and Stress 9.3. Easy Bijections Some of the bijections associated with 09 can now be confirmed. T h e o r e m 20, Let P be as above. Then Cod: SJ ~ S~ is a bijection. Further, if d > 3, then wd-2: Se_l ~ Sel is a bijection. Proof The first statement is trivially true by part (1) o f the previous corollary simply because P* has positive (and hence nonzero) volume. From the D e h n Sommerville relations we know that dim Sde_l = hd-i = hi = dim S(. So it suffices to show that 09d-2:S~_1 ~ S~ is a surjection. F r o m T h e o r e m 16 we know that { r l ( V ( x ) ) . . . . . rn(V(x))} spans SJ_ 1. We need to show {09d-2r1(V(x)) . . . . . wd-2rn (V (x))} spans S~. Since dim S~ = h l = n - d, it is sufficient to demonstrate that the given subset of S~ has rank n - d . However, since 09and ri commute, this subset equals ( d - 2 ) ! {rl (W2 (x)) . . . . . r, (W2 (x)) }. It is straightforward to check that )--~__-1xi ri (W2 (x)) = 2W2(x). It is known from the B r u n n - M i n k o w s k i theory that the quadratic f o r m W2(x) has d zero eigenvalues (associated with the space of translations x = (u r Vl . . . . . u r v,)), one positive eigenvalue, and n - d - 1 negative eigenvaiues; see, for example, [ 5 ]. So the rank of the quadratic form is n - d, as is required. Note that this is the case r = 1 of the H o d g e - R i e m m a n n - M i n k o w s k i inequalities developed by McMullen [ 13 ]. [] This theorem implies that 09: Se ~ S( is a surjection when d > 3. Therefore dim S~ = h2 - hi = g2 > 0, and we have bound together the main ideas of Stanley's and Kalai's different proofs of the Lower Bound Theorem. 9.4. Simplicial 3-Polytopes What we have done so far essentially gives a complete description of the situation for simplicial 3-polytopes. If P above is three-dimensional, then: 1. The canonical linear 0-stress is 3 ! times the volume of P*. 2. The canonical linear 1-stress is equivalent to the linear relation induced by Minkowski's theorem. 3. The canonical linear 2-stress is the classical Maxwell stress (shown by FiUiman [ 5 ]). 41 S e = R . 5. Sf has dimension n - 3 and is isomorphic to the space of all linear relations on the vi. 6. $2e has dimension n - 3, is spanned by the r i V ( x ) , and is isomorphic to S~ under the bijection induced by multiplication by 09. 7. $3e is s p a n n e d b y V ( X ) . 8. S ~ = R . 9. S~ has dimension n - 4 and is isomorphic to the space of all affine relations on the vi. 10. S~ and S~ are trivial. 11. That o93:$3e --+ Sot is a bijection is equivalent to P* having nonzero volume. 12. That o9:$2e ~ S~ is a bijection is equivalent to infinitesimal rigidity of the edgeskeleton of P and and is a consequence of the Brunn-Minkowski theory. So already in dimension three there is a striking confluence ofgeometric and algebraic results. Relationship to Weights 10.1. Ring o f Differential Operators Let A be the boundary complex of a simplicial convex d-polytope P containing the origin in its interior, and let Vl. . . . . vn be the vertices of P. Consider the ring R[O/OXl . . . . . O/OXn] of all differential operators with constant coefficients in the variables xl . . . . . xn. Define the polynomial V ( x ) as before and factor out the ideal of operators that annihilate the polynomial V ( x ) . Khovanskii (personal communication) observes that the resulting ring D is isomorphic to the cohomology ring of the projective toric variety associated with P. This implies the following result, which can be proved directly. Theorem 21. Let A be as above. Then D is isomorphic to B = A/(01 . . . . . Od) where the coefficients o f the Oi are related to the coefficients o f the vertices o f P as in Section 3.2. Proof. Clearly, r s ( V ( x ) ) equals zero for any subset S r A. However, the invariance of the polynomial V (x) under translation (see the proof of Theorem 17) implies that a v ( x ) i=l 1)ij OX-------Z _ 0 for each j = 1. . . . . d. Finally, Theorem 16 implies that the image of V ( x ) under the homogeneous differential operators of degree k spans Sde_~,hence has dimension hd-k ----hk. Using Theorem 2, this suffices to prove that D is isomorphic to B under the map O/Oxi --~ X i . [] This viewpoint allows us to define a multiplication on stresses. Let a ( x ) and b ( x ) be linear stresses. Find operators a' and b' such that a' ( V (x ) ) = a (x ) and b' ( V (x ) ) = b(x ). Define a ( x ) . b(x) to be the linear stress ( a ' b ' ) ( V ( x ) ) . The multiplication is well defined, f o r i f a " ( V ( x ) ) = a ( x ) and b " ( V ( x ) ) = b(x), then (a"b" - a'b') (V (x)) = (a"b" - a'b" § a'b" - a'b') (V (x)) = b"(a" - a ' ) ( V ( x ) ) + a'(b" - b ' ) ( V ( x ) ) ~ - 0 . It is also clear that the product o f a (d - / ) - s t r e s s and a (d - j ) - s t r e s s is a (d - i - j ) stress. Writing ~/e = S~_i" we then regard the space o f all linear stresses as a graded algebra S0e <9--- ~ Sae.This algebra is isomorphic to D, hence also to B. It sometimes helps to take a slightly schizophrenic viewpoint, on the one hand thinking o f a linear stress a(x) as a polynomial, and on the other identifying it with the operator a ' for which a'(V (x)) = a(x). 10.2. Weights McMullen reproved the g-theorem using the notion of weights on polytopes. An i-weight on a convex polytope P is a real-valued function a on t h e / - f a c e s of P which satisfies the Minkowski relation Z a(F)UF, G = 0 FcG for each (i + D-face G o f P. Here the sum is taken over all i-faces F contained in G, and UF,G is the unit outer normal vector o f F with respect to G within a f f G . Clearly, one n a t u r a l / - w e i g h t is given b y a ( F ) = v o l i ( F ) for each i-face F. We call this the canonical i-weight. The real vector space o f / - w e i g h t s on P is denoted ~ i ( P ) , and we denote ( ~ = o ~ i ( P ) by ~ ( P ) . McMullen [ 14 ] defines a multiplication on f ~ ( P ) that endows ~ ( P ) with a graded algebra structure. T h e o r e m 22. Let A be the boundary complex of a simplicial convex d-polytope P C R d containing the origin in its interior, and take Vl . . . . . vn to be the vertices of P. Then S/e(A) is isomorphic to f2e-i( P*) as vector spaces. Proof. Suppose that b(x) is a linear k-stress. Then, by condition (6), for every face G o f cardinality k - 1 we have the condition Z i~lk G bG+iWi = O. For a particular i ~ lk G and F --- G + i, wi is in the same direction as the corresponding uG*,r*. So, writing (G) for vold_k(conv({O} U {vi: i ~ G})) and similarly for (F), we have E F=G+iDG ~-~ bG+itOi : iElkG O, bFUG,.F*IIWill = O, E F~G bFUG*'F* (F) = O, (G) E F*cG* bFUG%F,(F) = O. Hence taking a(F*) = b F ( F ) for each (d - k)-face F* yields a (d - k)-weight on P*. [] Although we do not give the details here, it can be shown that the above map is an algebra isomorphism from S0~ ~ . - - ~ Sde tO f2(P*), and that the ring D of differential operators is isomorphic to the polytope subalgebra FI (P*) defined by McMullen [ 13 ]. a pi Let p = ~, and Pi = Wi, which corresponds to the canonical/-weight on P*. Then Pi = pi / i ! (keeping in mind our willingness to confuse at our convenience a differential operator with the image of V(x) under the action of the operator). Formally writing [P*] = ~_,di=lPi, we have This corresponds to the result of McMullen [ 13 ], [ 14 ] that [P*] = e x p p and p = log[P*]. 10.4. Restrictions The dual interpretation of Theorem 15 is interesting. Let G be a face of P of cardinality s, and let a(x) be the unique (up to scalar multiple) (d - s)-stress supported on clstar G. Then a(x) is dual to the necessarily unique s-weight a* supported on the set of all sfaces that meet G*. We call a* the weight associated with G*. In particular, the weight associated with a facet F* of P* is a 1-weight that is supported on the edges of P* that meet F*, and so this must be the same weight described by McMullen in [ 14 ]. What is dual to the notion of the restriction of a n / - w e i g h t a* on P* to a facet F*? Let a(x) be the dual (d - / ) - s t r e s s on P and let vk be the vertex o f P corresponding to F*. We could simply truncate a (x), eliminating the terms not supported on openstar Vk. This would not necessarily be a stress on openstar Vk, but it would directly correspond to the restriction of a* to F*. On the other hand, we can apply O/Oxk to a(x), which depends only on the terms of a(x) that are supported on openstar Vk. This gives a (d - i - 1)stress on clstar vk. Projecting clstar vk onto a hyperplane orthogonal to vk, deleting Ok,and applying Theorem 7 yields a simplicial (d -- 1)-polytope F dual to F* and a (d - i - 1)stress dual to the restriction o f a* to F*. We now have another way of interpreting McMullen's alternative formula [ 13 ], [ 14 ] for multiplying by a 1-weight a*. Remembering that we can view a linear i-stress on P as either a polynomial o f degree i or an operator o f degree (d - i) as it suits us, we choose to let a* correspond to an operator a = )--~=1 rlj(O/Oxj) o f degree 1, and to let any other given weight y* correspond to a polynomial y. Then which can be regarded as a dual recasting o f McMullen's formula n ya = E i=1 Oy rlj Oxi' n where a*(F*) denotes the restriction of a* to the facet F*. 10.5. Shellings and Flips McMullen [ 13 ] uses a shelling argument directly on the simple polytope P* to find a basis for f2 and to prove that dim f2i = hi, i ----0 . . . . . d. A general hyperplane is moved "upward" through P*. When a vertex of type i is encountered (i.e., a vertex with exactly i edges "below" the hyperplane), an arbitrary/-weight can be assigned to t h e / - f a c e F determined by these edges. The Minkowski relations defining the weights can be used to find the unique/-weights that must be assigned to t h e / - f a c e s that do not have a vertex o f type i at the top. It happens that this basis is not dual to the closed star stress basis constructed in Section 8. For the dual basis, again a general hyperplane is moved upward through P*. W h e n a vertex o f type i is encountered, we take F* to be the a c c o m p a n y i n g / - f a c e and add into the basis for f2d_i the (d -/)-weight associated with F* as in the previous section. In some sense the elements of this basis are more local than those o f McMullen's basis. We conclude with some comments on McMullen's flips [ 13 ]. As he points out, flips are dual to bistellar operations. Even though we are considering bistellar operations in a more general context, it is straightforward to verify that our Theorem 13 is dual to McMullen's Theorem 11.3 in [ 13 ], and that the justifications of these two theorems are essentially the same in a combinatorial sense. 11. In Section 5 we considered a matrix whose left nullspace defined affine k-stress. The fight nullspace then turned out to be an appropriate generalization of infinitesimal motions. We can try the same procedure with k-weights on a simple d-polytope P. For each (k + 1)-face G, consider a rigid motion ~oGthat maps affG onto (R k+l , 0 . . . . . 0) C R d and then projects this space naturally onto R k+l. Construct a matrix R with one row for each k-face F of P and columns occurring in groups of k + 1, one group for each (k + 1)-face G. The row vector of length k + 1 in row F, group G, is O r goG(u~,a) if if F f f . G, F C G, where UF,6 is the unit outer normal vector of F with respect to G in affG. Define m = ~0-1 (m') (where ~o-1 is interpreted in the obvious way) to be a (k + 1)circulation when m' is a member of the fight nullspace of R. So m is an assignment to each (k + 1)-face G of a vector parallel to G that satisfies the conditions for every k-face F. Denote the space of (k + 1)-circulations by Ck+l. In the case that k = 0, we have a vector, or flow, associated with each edge of P and a condition on each vertex of P that forces flow conservation. For higher values of k we can interpret the (k + 1)-circulation as a translation of the (k + 1)-dimensional content of the (k + 1)-dimensional faces in directions parallel to these faces with flow conservation across every bounding k-face. Theorem 23. Let P be a simple d-polytope. Then dim C k + 1 -~- hk -- f k -I- (k + 1)fk+l. Proof. This is an immediate consequence of the fact that R is an fk x (k + 1)fk+l matrix with a left nullspace of dimension hk. [] It is clear that dim C1 = h0 - f0 + fl = fl - fo + 1, which is the dimension of the space of ordinary circulations on a graph with f0 vertices and fl edges. In general, in terms of the f - v e c t o r of the simple d-polytope P (the reverse of the f - v e c t o r of the dual simplicial polytope) So h * = Ed ( _ 1)j-k(J)k 3~ j=k = f k - - ( k + l ) f k + l + E j=k+2 ( _ l ) J - k J J-J" dim Ck+1 d E j=k+2 ( k,k] j" 12. Unbounded Simple Polyhedra Consider the boundary complex A of a simplicial convex d-polytope P with vertex set vl . . . . . vn. Consider the ring B = Bo <9... <9 Ba = A/(O1 . . . . . 0n), where the Oi are constructed from the vj as in Section 3.2. As we have already mentioned, the g-theorem implies that hi = hd-i, i = 0 . . . . . ld/2J and also that gi > gi-1, i = 1. . . . . / d / 2 J , and this is proved by showing that multiplication by wd-2i is a bijection between Bi and Bd_i, i = 0 . . . . . /d/2J. Now let v be any vertex of P and consider the simplicial complex E = A \ v . In [ 3 ] it is proved that h i ( E ) > hd_i(E), i = 0 . . . . . [ d / 2 J , and also that h i ( E ) > h i + l ( E ) , i = Ld/2J . . . . . d. This is a consequence of the g-theorem, but now we can view this as a consequence o f a weakened Lefschetz-type theorem on the face ring o f E. Let At = R[xl . . . . . xn]/l:: and B' = B~ < 9 . . . <9 B'd = A'/(O1 . . . . . Od). Take o: = Xl + . . . + xn as before. Theorem 24. Multiplication by (0d-2i is a surjectionfrom Bi to Ba-i, i = 0 . . . . . / d / 2 J . It is more convenient to prove this with weights instead o f stress. Let P* be the simple d-polytope dual to P and let F* be the facet o f P* corresponding to v. It can be arranged (for example, by choosing the origin suitably close to v) that discarding the inequality defining the facet F* results in an unbounded simple polyhedron Q* which is dual to the simplicial complex E. We can define weights on Q* in the natural way, even though Q* is unbounded. So dim [2i(Q* ) = h d - i ( E ) , i = 0 . . . . . d. What we actually prove is: Theorem 25. 0 . . . . . l a / 2 j . Multiplication by pd-2i is an injection f r o m ~'~i(Q*) to ~'2d_i(Q*), i = Proof. Use McMullen's construction to consider a basis o f f2(P*) determined by a hyperplane. Choose this hyperplane so that it first moves past the vertices in F* before it encounters the remaining vertices of P*; i.e., arrange for F* to be at the "bottom" of P*. Consider a vertex vj o f type i in P and the associated basis element a o f f2i (P*). In the case that vj is also in F*, vj is also a vertex of type i in F* and the restriction o f a to F* is an element o f the basis o f f2 (F*). Reversing this restriction gives an injection o f f2i (F*) into ~'2i (P*). In the case that vj is not in F*, the restriction o f a to F* is zero. Notice that the weights o f Q* correspond naturally to the weights o f P* that are zero on F*. Hence we have an injection of f2i (Q*) into ~i(P*). Identifying f2i (F*) and f2i (Q*) with their images in [2i(P*) yields the direct sum decomposition ~ i ( P * ) = ~2i(F*) <9~i(O*). Looking at the description of the multiplication of weights in McMullen [ 13 ], it is seen that multiplying p by a weight that is zero on F* results in a weight that is also zero on F*. Therefore, since multiplication by pd-2i is a bijection from f2i ( P ) = f2i (F*) <9 ff2i(Q*) to ~ d - i ( P ) = ~ d - i ( F*) <9 f2d-i(Q*), it must be an injection from ~2i(a* ) to ~'~d-i ( Q*). [] Acknowledgments Many colleagues and agencies have provided me with both material and intellectual stimulation and support for this work, which has been evolving over a period o f years. In particular, the investigations which eventually led to the definition o f k-stress were motivated by Kalai's results on algebraic shifting and on stress. Most o f the results in Sections 3, 4, 6 - 8 were developed in 1987-1988, but the basis for the formulas for the coefficients rests in some calculations begun during the author's stay at the Mathematical Institute of the Ruhr University in Bochum, 1984-1985, which was supported by a fellowship from the Alexander von Humboldt Foundation. Jonathan Fine's suggestion in Oberwolfach 1989 that the volumes o f the faces of the dual simple polytope should play an important role led to the material in Section 9, mostly discovered during the author's stay at DIMACS, Rutgers University in 1989-1990. The definition o f infinitesimal motions (Section 5) resulted from conversations with Nell White during a visit to the Mittag-Leffler Institute in early 1992. 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Stanley , The upper bound conjecture and Cohen-Macaulay rings , Stud. Appl. Math., 54 ( 1975 ), 135 - 142 . 20. R. P. Stanley , Hilbert functions of graded algebras, Adv . in Math., 28 ( 1978 ), 57 - 83 . 21. R. P. Stanley , The number of faces of a simplicial convex polytope, Adv . in Math., 35 ( 1980 ), 236 - 238 . 22. T .-S. Tay, N. White , and W. Whiteley , Skeletal rigidity of simplicial complexes, I, II , European J. Combin ., 16 ( 1995 ), 381 - 403 , 503 - 523 . 23. T .-S. Tay, N. white, and W. Whiteley , A homological interpretation of skeletal rigidity , Manuscript , 1993 .


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C. W. Lee. P.L.-Spheres, convex polytopes, and stress, Discrete & Computational Geometry, 1996, 389-421, DOI: 10.1007/BF02711516