Discrete Comput Geom 11:465~J,76 (1994) Discrete & Computational Geometry

�9 1994 Springer-Vertag New York Inc.

O n Subdiv i s ions o f S i m p l i c i a l C o m p l e x e s :

Characterizing L o c a l h - V e c t o r s *

C. Chan

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA clara@ math.mit.edu

Abstract. A well-known combinatorial invariant of simplicial complexes is the h-vector, which has been thc subject of much combinatorial research. This paper deals with local h-vectors, recently defined by Stanley as a tool for studying h-vectors of simplicial subdivisions. The face-vector of any simplicial complex can only increase when the complex is subdivided; how does the h-vector change? Motivated by this question, Stanley derived certain useful properties of local h-vectors. In this paper we use mainly geometric arguments to show that these properties characterize local h-vectors, and regular local h-vectors.

1. Preliminaries

For general background and terminology on combinatorics of simplicial com- plexes, see Chapter 2 of [$2] or Sections 1.2 and 2.2 of [C-I.

Before we begin, we need some notation:

For any n e Z, [n] denotes the set {1, 2 . . . . . n}. For any finite set V, # V denotes the number of elements in 1/, and 2 r denotes

the simplex on vertex set V, {W: W ___ V}. For any simplicial complex A, IAI denotes the geometric realization of A, and

if F is any face of A, then IFI denotes the image of F in IAI. (See Section 0.3 of [$2]. The link o f f in A, denoted by lkAF, is the subcomplex

{G~A: G w FeA, G c~F = ~} .

* An earlier version of this work appeared in the author's Ph.D. thesis at the Massachusetts Institute of Technology, June 1992.

466 c. Chan

For any (disjoint) simplicial complexes A and F, A . F denotes the simplicial complex {F u G: F e A, G e F}, which is called their simplicial join.

Now let A be a finite (d - 1)-dimensional simplicial complex with f~ i-dimen- sional faces (/-faces) for each i. Then the d-tuple (f0 . . . . . fd- 1) is called the f-vector of A, denoted f(A). By convention, A has one ( - 1)-dimensional face (the empty face), so f -1 = 1, unless A = ~ (in which case f -1 = 0).

An algebraically desirable alternative to the f-vector is the h-vector h(A):-- (h o . . . . . hal), defined by

d d

E f / - l ( x - 1) a - i = ~ hl xa-i" i = 0 i = 0

d In particular, the h-polynomial h(A, x) = ~i=o hi xi is algebraically convenient to work with. For example, h(2 v, x) = 1 for any simplex 2 V. Also, for any disjoint simplicial complexes A and F, we can easily compute the h-vector of their simplicial join

h(A * F, x) = h(A, x). h(F, x).

From the definition, it is easy to derive the useful identity

h(A, x) = ~ x 'F (1 -- X) a -*e . F e A

See [$3] for background on the algebraic significance of h-vectors, and what has been done to characterize h-vectors of large classes of simplicial complexes. Some of this characterization utilizes results from homological algebra and intersection homology theory.

Of particular interest are the C o h e n - M a c a u l a y complexes, as defined, for example, in [S1]. These simplicial complexes have been the subject of much research in both algebraic and topological contexts [$2, Section 2.4], [Re]. For example, all simplicial spheres and balls are Cohen-Macaulay. It is known that h-vectors of Cohen-Macaulay complexes are nonnegative, and in fact they have been completely characterized [$2, Section 2.3].

2. Subdivisions of Simplieial Complexes

A natural question to ask about h(A) is whether or not it increases, as f(A) does, when some faces of A are subdivided, i.e., if h(A) = (ho, h 1 . . . . . ha) and h(A') = (h~, h'l . . . . . h~), where A' is a subdivision of A, then must h'i > hi hold for each i?

For example, if A is the boundary of a tetrahedron, and A' is the subdivision of A pictured in Fig. 1, then the answer is yes. In fact, if A is any Cohen-Macaulay complex, the answer is yes, at least if the subdivision is quasi-geometric (defined

On Subdivisions of Simplicial Complexes: Characterizing Local h-Vectors

1

(a) (b)

Fig. 1. (a) h(A) = (1, 1, 1, 1); (b) h(A') : (1, 2, 2, 1).

467

below). The p roof uses certain propert ies of local h-vectors of the subdivision restricted to each face of A [$4].

Before moving on, let us formally define subdivision and describe three basic kinds of subdivisions (first defined in [$4]).

D e f i n i t i o n . Let F be a simplicial complex and let a be a m a p from F to the simplex 2 v such that:

(i) For each W ~_ V, the restriction Fw = a - 1(2w) is a subcomplex of F with geometr ic realization homeomorph ic to the ball B * w - 1.

(ii) For each F e F, a(F) = W if and only if F is an interior face of Fw, i.e., IFI is not contained in the boundary of IFwl.

Then F is called a subdivision o f 2 v with subdivision map a. For each F ~ F, we call a(F) the carrier of F, and say F lies on W if a(F) c I4.

Let A and A' be simplicial complexes and let a be a m a p from A' to A. For each face F e A , let A~ denote a- l(2v) . Then if, for each F, cr restricts to a subdivision m a p for A~ as a subdivision of 2 r, we say A' is a subdivision o f A with subdivision map a.

It is convenient to be able to picture a subdivision of 2 v as a realization in R n in the following sense.

D e f i n i t i o n . If F is a subdivision of 2 v with subdivision map a, then a realization of (F, a) is a geometric realization of IF[ in R n such that, for each W _ V, ]Fw[ is the convex hull of {1 a - X({x})l: x ~ W}.

Example. Let F have maximal faces {1, 2, 4}, {2, 3, 4} and let a: F --, 2 TM be given by a({4}) = a({1, 4}) = a({3, 4}) = {1, 3} and a({2, 4}) = a({1, 2, 4}) = a({2, 3, 4}) = {1, 2, 3}, with a(F) = F for all other F e F. Then a realization of F is shown in Fig. 2.

Basic Kinds o f Subdivisions

A subdivision F of 2 v is geometric if it can be realized in R d with all convex faces. The subdivision in Fig. 2 is geometric, for example.

468 C. Chan

2 3

Fig. 2.

A subdivision F is quasi-geometric if no face F �9 F has all its vertices lying on a face of 2 v of dimension less than dim(F). Every geometr ic subdivision is clearly quasi-geometric, but the converse is not true. D o u g Jungreis has suggested a simple example:

Start with a te t rahedron and cut away small convex te t rahedra leaving only a knot ted tube, as in Fig. 3(a). Finally, put a new vertex inside the tube and join it with the bounda ry faces of the tube. This is quasi-geometric, but not geometric. On the other hand, not every subdivision is quasi-geometric. See Fig. 3(b).

A geometr ic subdivision F of 2 v is regular if it is the project ion of a strictly convex polyhedral surface (with boundary) in R d+l. Formal ly , F is regular if a height function ~o:1FI ~ R which is piecewise linear and strictly convex exists, i.e.:

(i) co is piecewise linear: For all F �9 F, m restricts to a linear function ~o v on IFI. (ii) o is convex: For all x, y e I F I , and all # � 9 1],

o (#x + (1 - /~)y) > p~o(x) + (1 - #)co(y).

(iii) For all F, G distinct maximal faces of F, the functions ~or, co o are distinct (as linear functions).

For example, let F be the subdivision of 2 v with one interior vertex z and maximal faces of the form F u {z}, where F is any (d - 1)-dimensional face of 2 v (see Fig. 4). Then F is a regular subdivision of 2 v.

Notes. (1) Every regular subdivision is shellable [BM]. Thus geometr ic subdivisions are not necessarily regular [Ru].

(2) Let A be a finite affine point configuration, and let ~ be the polytope conv(A), whose vertices are contained in A. Then the regular subdivisions of

are in one- to-one correspondence with the vertices of the secondary polytope Z(A) [BFS].

2 ~ 4 2 (b) (a)

Fig. 3

On Subdivisions of Simplicial Complexes: Characterizing Local h-Vectors 469

Fig. 4. F is a projection of the tent.

3. Local h-Vectors

Let V = [d] for some nonnegative integer d. We now give the definition and key properties of the local h-vector of any subdivision of the simplex 2 v. See [$4] for elaboration and proofs.

Definition. Let F be a subdivision of 2 v. Its local h-polynomial is defined by

lv ( r ,x )= ~ ( - 1 ) * v - * W h ( F w , x), W c _ V

and its local h-vector Iv(F) is (lo . . . . . la), where Iv(F, x) = ~.f=o lixk See Lemmas 4.2--4.4 for examples.

The following result validates the terminology local h-vector.

Theorem 3.1 [$4, Theorem 3.2]. I f A' is any subdivision of a pure simplicial complex A, we have

h(A', x) = ~ lr(A'F, x) 'h(lkaF, x). F e A

Theorem 3.2 [$4, Corollary 4.7]. I f F is quasi-geometric, then lr(F ) is nonnegative, i.e., li > O, for all 0 < i <_ d.

Theorems 3.1 and 3.2 result in a partial answer to the motivating question behind local h-vectors.

Corollary 3.3 [$4, Theorem 4.10]. I f A' is a quasi-geometric subdivision of a Cohen-Macaulay simplicial complex A, then h(A') > h(A).

We are particularly interested in the following properties of local h-vectors.

Theorem 3.4 I-$4, Theorems 2.3, 3.3, 5.2]. The local h-vector of any subdivision is symmetric, i.e., I i = Id_ i for 0 < i < d. Furthermore, lo = 0 and Ix > O. I f the subdivision is regular, then lv(F ) is unimodal, i.e.,

1o ~ 11 <_ .. . < lkd/2 j >-- . . . > l a .

470 C. Chan

4. Characterization of Local h-Vectors

Let V = [d], where d is a positive integer. We now show that local h-vectors of arbitrary subdivisions of 2 v can be completely characterized as follows:

Theorem 4.1 [C, Theorem 2.4.1]. Le t I = (lo . . . . . la)e Z a+ l. Then 1= Iv(F) for some subdivision F o f 2 v i f and only i f l is symmetric and l o = O, l I >_ O.

As mentioned in Section 3, the "only if" direction has already been proved. To prove the "if" direction, we construct a subdivision F, given any l satisfying the above requirements, so that /v(F) = I. We can do so with the help of three lemmas.

Let F be a subdivision of 2 v. See Figs. 5-7 for illustrations of the subdivisions described in the lemmas below.

Lemma 4.2. Let F be a maximal face o f F, and let F' be the subdivision o f F

with one new vertex z in Int(F). So the face F o f f is replaced by the faces f w {z} in F'. Then h(F', x) = h(F, x) + x + x 2 + "'" + x d- 1, and

Iv(F', x) = Iv(F, x) + x + x z + . . . + x a- 1

Proof. We first show that h(F', x) = h(F, x) + x + X 2 "q- " ' " + X d - 1 :

h(r ' , x) = ~ x*~ - x) a - * ~ G e F '

G ~ F , G ~ F G ~ F

X = h(F, x) - x 'V (1 - x) a - * e + ~ - x ~ x'G(1 - x)a- *~

G=F

X = h(F, x) - x*V(1 - x) d- . v +

1 - x (h(2 r, x) - x~r(1 - x) a- ,~v)

x = h(F, x ) - x a + (1 - x a)

1 - x

= h(F, x) + x + x 2 + " " + x a-1

Fig. 5

On Subdivisions of Simplicial Complexes: Characterizing Local h-Vectors

Fig. 6.

471

A copy of G is pushed into the interior of F, as in the two-dimensional case of Fig. 3(b).

N o w we show tha t l v (F , x) = / v ( F , x) + x + x 2 + " . + x a - 1. Since F~v = Fw for all W r V,

I v ( r ' , x ) = ~ ( - l ) d - e W h ( F ' w , X ) W~_g

= h(F', x) + ~, ( - l ) d- *Wh(Fw, x) W = V

= h(F' , x) + Iv(F, x) - h(F, x)

= lv(F, x) + x + x z + "" + x d-1. []

L e m m a 4.3. Let d > 4. Le t G be a (d - 2)-dimensional face o f F with (d - 2)- dimensional carrier W, and let F' be the subdivision o f f with one new ver tex w (with carrier W) and one new max imal f ace G w {w). So in F', G has carrier V. Then h(F' , x) = h(F, x) + x, and lv(F', x) = lv(F, x) - x 2 . . . . . x d- 2.

P r o o f W e first show tha t h(F' , x) = h(F, x) + x:

h(F' , x) = ~, x ' F ( 1 -- X) a- ~F FeY '

= E X~r(1 -- X)a-*l: + L X*rU~w~( 1 -- X)a-*ruIw) F ~ F F ~ G

= h(r , x) + x" ~ x#F(1 - x) d - l - # F Fe 2 ~

= h(F, x) + x" h(2 ~, x)

= h(F, x) + x.

Fig. 7. 1 ~ is six tetrahedra glued together.

472 C. C h a n

Now we show that /v(F ' , x) = Iv(F, x ) - x 2 . . . . . x d- 2. Since F~; = F v if U N W,

lv(r ' , x) = ~ ( - 1) d- •Vh(Fb, x) U ~ V

= h(r ' , x) - h(r~,, x) +

(applying L e m m a 4.2 to Fk)

~, ( - - 1) a- *Vh(Fv, x) UNW

= h(F', x) - h (Fw, x) - (x + " " + x a- 2) + lv(F ' x) - h(F, x) + h(Fw, x)

= h(F', x) - (x + x 2 + "'" "~- X d - 2) ..~ lv(F ' x) -- h(F, x)

= Iv(F, x) - x 2 . . . . . x d- 2. []

L e m m a 4.4. L e t f~ be the subdivision o f 2 {d+ l"d+ 2} with one interior vertex. Then

/f 1~ = [d + 2] and ~ = F �9 ~, we have Ir x) = x "Iv(F, x).

P r o o f

Igr', x) = Y~

= ( - -1 ) d

= ( - -1 ) d

( - 1) a+z - * ~Vh(l~, x )

( - 1 ) * ~Vh(~v, x)

~ ( - - 1)*WuVh(~wuv, x). Wc-V U~{d+l ,d+2}

For each W__ V a n d U _ _ { d + 1, d + 2}, we have

h(~'w~ v, x) = h((F w �9 ~v), x) = h(F w, x ) . h(f~v, x).

I t is easy to compute

h(f~ o , x) = h(f~d+l>, x) = h( f~§ x) = 1,

So

1r ~, x) = (-- 1) a

- - x " ~ ( - 1 ) d - * w h ( F w , x ) Wc_V

= x " I v ( F , x ) .

N o w we can put it all together.

h([~2{d+ l ,a§ 2), x ) = 1 -b x .

( - 1)*W{h(Fw, x) - h(F w, x) - h(Fw, x) + (1 + x) " h(Fw, x)} W---V

[ ]

On Subdivisions of Simplicial Complexes: Characterizing Local h-Vectors

Fig. 8

473

P r o o f o f Theorem 4.1. Given lo = 0, I t > 0, and li = ld-i for all i, we want a subdivision F of 2 v such that

d

Iv(F, x) = l(x) = ~ l lx i. i = 0

If d < 3, we can construct F directly from 2 v by l~ applications of L e m m a 4.2. Fo r example, we can get l = (0, 2, 2, 0) by putt ing a new vertex x in the face {1, 2, 3} of 2 t31, and then putting a new vertex y in the face {1, 2, x} of the resulting subdivision (see Fig. 8).

Analogously, for d > 4, if we have a subdivision F0 of 2 v with

lv(Fo, x) = l(x) -- Ii " (x + x 2 + "'" + x a - l ) ,

then we can get F from F o by 11 applications of Lemma 4.2. We now show that such a subdivision F 0 can be constructed. Let s = 12 - ll.

We can assume by induction that a subdivision F ' of 2 [d-2l exists with

l[a 210-",X)= X - I ' ( I ( x ) - - I I ' ( x + X 2 + " " + X a - 1 ) - s ' ( X 2 + ' " + X a-z)).

If s < 0, apply L e m m a 4.4 to get a subdivision of 2 v f rom F', and then get F o by Is] applications of L e m m a 4.3. If s > 0, start with F', apply L e m m a 4.2 s times, and then apply L e m m a 4.4 to the resulting subdivision to get F o.

For example, let d = 4 and l = (0, 1, - 1, 1, 0), see Fig. 9. First we form the simplicial join 2 t21 �9 f~, as in L e m m a 4.4. Then we put a vertex x in a 2-face on the bounday, and push the original 2-face into the interior, as in L e m m a 4.3. Since s = 12 - 11 = 2, we do this operat ion again (with another vertex y) to get Fo. Finally we put a vertex in a maximal face of F o (as in L e m m a 4.2) to get a subdivision F of 2 v with the desired local h-vector. [ ]

3 / ~ 3 F

1 1 2 @ Fo ~ 2 1 4 I ~. j, 1 , ~ - - - - ~ \ [ ~ -4 ~" 4

Fig. 9

474 C. Chan

Fig. lO

X

5. Local h-Vectors of Regular Subdivisions

Using the same construct ion as in the previous section, we now show that local h-vectors of regular subdivisions of 2 v can be characterized as follows:

Theorem 5.1 [C, Theo rem 2.5.1]. l f l = (lo, l l , . . . , l~) ~ Z n, then l = lv(F) for some regular subdivision F o f 2 v i f and only if lo = 0 and l is unimodal and symmetric.

Proof. Stanley p roved the "on ly if" direction in [$4]. To prove the "if" direction, it suffices to show tha t the subdivision F constructed in the proof of Theorem 4.1 is regular when lo --- 0 and l is unimodal and symmetric. F o r V = [d], let F be a regular subdivision of 2 v and let ~o: ]FI --* R be piecewise linear and strictly convex.

L e m m a 5.2. F' = F �9 {x} is a regular subdivision o f 2 vU~x~ (see Fig. I0).

Proof. Let ~o' be the unique piecewise linear extension of ~o to F ' such that ~o'(x) = 0. I t is easy to verify tha t o9' is strictly convex. So ~o' is a height function for F'. [ ]

L e m m a 5.3. Let G be an i-face of F with i > 2, and let F' be the subdivision of F which results from putting a new vertex v in the interior of G and joining v with all

faces of the form G' -- {z}, where G' ~ G and z ~ G. Then F' is regular (see Fig. 11).

Proof. Based on the corresponding stellar subdivision of the polyhedron bounded by the g raph of ~o, it is easy to define a height function co': F ' ~ R. [ ]

(a) (b)

Fig. 11. (a) Two tetrahedra intersecting on G. (b) Six tetrahedra.

On Subdivisions of Simplicial Complexes: Characterizing Local h-Vectors 475

3

1I t ~ 1 ,4 ~ 1 4 2

2 2

Fig. 12

Proof o f Theorem 5.1. As mentioned earlier, we only need to show the "if" direction. Given l ~ Z d+l symmetric and unimodal with lo = 0 , let F be the subdivision of 2 v with lv(F) = l, constructed as in the proof of Theorem 4.1. Since l is unimodal, s > 0 at every step of the construction. So F is built up only from subdivisions described in Lemmas 4.2 and 4.4.

For example, let d = 4 and l = (0, 1, 2, 2, 1, 0), see Fig. 12. As in the example at the end of Section 4, we start with 2 t2l and do the operations described in Lemma 4.2 and then in Lemma 4.4 (once each) to get F o. Then we do the operation described in Lemma 4.2 once more to get F with the desired local h-vector.

The subdivision in Lemma 4.2 is achieved by putting one new vertex in a (d - 1)-face as in Lemma 5.3. The subdivision in Lemma 4.4 is achieved by joining a given simplicial complex with a vertex x as in Lemma 5.2, and joining the resulting complex with another vertex y as in Lemma 5.2, and then putt ing a new vertex z in the face {x, y} as in Lemma 5.3. Thus, since we begin the whole process with a trivial subdivision 2 w for some W, our construction remains regular at each step (by Lemmas 5.2 and 5.3), so the resulting subdivision F is regular. [ ]

6. C o n c l u s i o n .

Thus we have completely characterized local h-vectors of subdivisions, and of regular subdivisions. An obvious next goal would be to characterize local h-vectors of quasi-geometric subdivisions.

C o n j e c t u r e [$4, Conjecture 5.4]. I f l = Iv(F) for some quasi-geometric subdivision F of 2 v, then l is unimodal.

If this conjecture is valid then our work in Section 5 shows that 1 o = 0 and l is symmetric and unimodal /f and only if 1 is the local h-vector of some quasi- geometric subdivision.

R e f e r e n c e s

[BFS] L. Billera, P. Filliman, and B. Sturmfels, Constructions and Complexity of Secondary Poly- topes, Adv. in Math. 83 (1990), 155-179.

476 C. Chan

IBM]

[c]

[Re] [au]

Is1]

[$2]

is3]

[$4]

M. Brugesser and P. Mani, Shellable Decompositions of Cells and Spheres, Math. Scand. 25 (1971), 197-205. C. Chan, On Shellings and Subdivisions of Convex Polytopes, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, June 1992. G. Reisner, Cohen-Macaulay Quotients of Polynomial Rings, Adv. in Math. 21 (1976), 3ff49. M. E. Rudin, An Unshellable Triangulation of a Tetrahedron, Bull. Amer. Math. Soc. 64 (1958), 9(~91. R. Stanley, Cohen-Macaulay Complexes, in Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht (1977), pp. 51-62. R. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, Vol. 41, Birkh~iuser, Boston (1983). R. Stanley, The Number of Faces of Simplicial Polytopes and Spheres, in Discrete Geometry and Convexity (J. E. Goodman et al., eds.), Annals of the New York Academy of Sciences, Vol. 440 (1985), pp. 212-223. R. Stanley, Subdivisions and Local h-Vectors, J. Amer. Math. Soc. 5(4) (1992), 805-851.

Received September 14, 1992, and in revised form November 25, 1992.