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BIGRADED BETTI NUMBERS OF CERTAIN SIMPLE POLYTOPES
IVAN LIMONCHENKO There is a well-known in toric topology construction which associates a smooth manifold with a combinatorial simple polytope. Topological and analytic properties of these toric varieties, called momentangle-manifolds, are in the area of main interest of toric topology and geometry. In this work we consider the manifolds, which arise from the parametried series of the so called Stashe polytopes and stacked polytopes. we compute bigraded Betti numbers in the case of Stashe polytopes Kn+2 with n 5 and the stacked polytopes of arbitrary dimension and an arbitrary amount of cut vertices. The bigraded Betti numbers in our examples are computed with the help of computer program Macaulay2, a brief description of the method is given below.
Abstract.

1.

Introduction

In the rst section we are going to introduce the notions of the so called face rings and bigraded Betti numbers of moment-angle-complexes that will be used during the whole work. The second section is dedicated to the case of the so called Stashe polytopes or associahedra. The third one is dedicated to the stacked polytopes, which also gives us a series of simple polytopes. The author is grateful to Taras Panov for fruitful discussions and advice which was always so kindly proposed during all this work. 2.

Stasheff polytopes

In the following we will denote by n the dimension, by m - the number of facets (vertices) of the simple (symplicial) polytopes. A few combinatorial ways to dene the Stashe polytope Kn+2 are known. For details, see [3, Lecture II]. We will point out the following

Theorem 2.1. (see [3, Lecture II, Theorem 5.1]). There exists an embedding
J :K
n+1

R

n-1

with an image
y = (y1 , · · · , y
n-1

) : 0 yl l · (n - l), yi - y

i+l

i · l,

where l = 1, · · · , n - 1, i = 1, · · · , n - l - 1. If one considers
Li,l = {y R
n-1

: yi - y

i+l

i · l, 1 l n - 1, 0 i n - l, y0 = yn = 0}.

stacked polytopes.

Key words and phrases.

Betti numbers, moment-angle-complexes, Stashe polytopes,
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IVAN LIMONCHENKO

than our description gives us the image of Kn+1 as an intersection of a (n - 1)-cube with all the sets Li,l . The proposition [3, Lecture II, Cor 6.2] allows us to reduce the computation of bigraded Betti numbers to the problem of nding the number of all noninteresecting pairs of diagonales in a corresponding polygon. In the examples below bigraded Betti numbers are got as a result of a program work, which is based on the following commands in the Macaulay2 computer stu:
k k = Z Z/32749, ring P = k k [v1 , · · · , vm ], I = ideal(· · · ), betti res(ring P 1 /I ).

in the rst line we generate the eld of coecients, in the third, in brackets, the monomial Stanley-Raisner ideal, consisting of elements of the form vi · vj is written, the res-command in the last line builds a minimal resolvent, and the betti-command gives us the cardinalities of the corresponding minimal bases. In what follows the result of the described program work when n 5 is given. For the convenience of the further observations, we will write the numbers appear in sets of (n - 1) vectors, which elements are the -i,2j , i = 1, · · · , m - n - 1, where j - i = 1, · · · , n - 1. Here we assume that

0 ,0

=

-(m-n),2m

= 1.

Example 2.2.

(1) n = 2, m = 5
(5, 5)

(2) n = 3, m = 9
(15, 35, 24, 3, 0) (0, 3, 24, 35, 15)

(3) n = 4, m = 14
(35, 140, 217, 154, 49, 7, 0, 0, 0) (0, 28, 266, 784, 1094, 784, 266, 28, 0) (0, 0, 0, 7, 49, 154, 217, 140, 35)

(4) n = 5, m = 20
(70, 420, 1089, 1544, 1300, 680, 226, 44, 4, 0, 0, 0, 0, 0) (0, 144, 1796, 8332, 20924, 32309, 32184, 20798, 8480, 2053, 264, 12, 0, 0) (0, 0, 12, 264, 2053, 8480, 20798, 32184, 32309, 20924, 8332, 1796, 144, 0) (0, 0, 0, 0, 0, 4, 44, 226, 680, 1300, 1544, 1089, 420, 70)


BIGRADED BETTI NUMBERS OF CERTAIN SIMPLE POLYTOPES

3

3.

Stacked polytopes

We are now to review the following

Denition. Lets call a stacked polytopeP (n, k) a simple polytope, which appears from a n-symplex by a consequtive cut of k vertices.
In our previous notation we immediately have k = m - n - 1. The dual symplicial polytope to the P (n, k) is called a pyramid suspension polytope Q(n, k). It can be obtained from the n-symplex by a consequtive suspension of cones above its facets. It is easy to convince oneself that for k = 1 and k = 2 the combinatorial type of the obtained polytope is independent of which vertices are cut. One can also easily see that for k 3 this does not remain true. Denote by Q Q(n, k), V - its vertex set, and the symplicial polytope Q is obtained from Q by only one suspension of a cone with a leading vertex v above its certain facet , which vertex set we will denote by V ( ). Lets also consider V - as a vertex set of Q , W V and KW - a full subcomplex on the vertex set of W in the sense of [1, p. 44]. Due to the well known Hochster's formula (see. [1, Theorem 3.17]) lets try to ask the following question: what is happening with the reduced homology groups of the full subcomplexes KW under some determined eld of coecients k when one goes from Q to Q ? Therefore we are going to consider the only 4 logically possible cases (in what follows we have W as a proper subset of V ) (1) v W, W V ( ) = If V ( ) W, then KW KW -{v} . If W V ( ) = V ( ), then = we have:
K
W -{v }

K

W V ( ){v }

= KW , K

W -{v }

K

W V ( ){v }

=K

W v ( )

.

But KW V () and KW V (){v} - in this case are, obviously, contractable. With the help of the Mayer-Vietoris exact sequence, we nally have:
~ Hi (KW , k ) Hi (K =~
W -{v }

, k ). {v }. From this

(2) v W, W V ( ) = In this case it is easy to see that K follows:
~ dimk Hi (KW , k ) = dimk dimk ~ Hi (K ~ Hi (K

W

=K

W -{v }

W -{v } W -{v }

, k ) + 1, for i = 0; , k ), for i > 0.

(3) v W, V ( ) W, W = V Consider a set of all subsets of V ( ). Then as a boundary of this set we will naturally have = - V ( ). It is clear that in this particular case we must obtain the following statements:
K
W

= KW , K

W

= ,


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IVAN LIMONCHENKO

(only the former facet F is vanishing.) But - is a symplicial (n - 2)-sphere, - is a symplicial (n - 1)-disk. Then with the help of the exact homology sequence with respect to i = (n - 1), one can soon obtain:
~ dimk Hi (KW , k ) = ~ dimk Hi (KW , k ), for i < (n - 2); ~ dimk Hi (KW , k ) + 1, for i = (n - 2).
W

(4) v W, V ( ) W In this case it is not hard to convince oneself, that K That's why we have:
~ ~ dimk Hi (KW , k ) = dimk Hi (KW , k ), i.

KW .

Now we can formulate our main statement as follows:

Theorem 3.1. For n 3 the fol lowing formulae for bigraded Betti numbers associated with the simple polytopes P (n, k) take place:

-i,2(i+1)

=i·

k+1 , i+1

-i,2(i+n-1)


-i,2(i+1)

-i,2j

k+1 , k+2-i = 0, j = i + 1, i + n - 1; = (k + 1 - i) ·

for n = 2 the fol lowing takes place:
=i· k+1 k+1 + (k + 1 - i) · , i+1 k+2-i
-i,2j

= 0, j = i + 1.

Proof. The rst formulae is proved in [4]; the second - its obvious consequence, if one notices the case of Poincare duality. The formulae above for the case of n = 2 - are obtained from the analogous formulae for grater n with the help of dimension matters (for example, nonzero element of the Betti number table - is just a sum of corresponding 2 elements nonzero elements from the case of higher dimension). Lest prove the rest of the statement about the zero Betti numbers. More precise:

Theorem 3.2. The fol lowing formula takes place:
~ Hi (KW , k ) = 0, i = 0, n - 2, = W V.

Proof. If m = n + 1, then P (n, k) - n-symplex, so KW - is contactable. The induction statement: lets take that the conclusion of the theorem is true for all proper subsets of W of V for all i = 0, n - 2, and consider W - as a proper subset of V . If W = V - {v } = V , then KW - is a symplicial (n - 1)-disk and that makes the conclusion be true. If W = {v }, then we can say the same. In the common case, according to what is mentioned above, while considering the 4 cases, we obtain the following:
~ ~ dimk Hi (KW , k ) = dimk Hi (K
W -{v }

, k ) = 0;


BIGRADED BETTI NUMBERS OF CERTAIN SIMPLE POLYTOPES

5

From our observations it is easy to see, that the following takes place

Corollary 3.3. Bigraded Betti numbers, associated with P (n, k) and its dual one Q(n, k), are independent of the combinatorial types of the latter ones, and the numbers, corresponding to j = i + 1 are as wel l independent of the dimension of the total euclidean space.
4.

Final remarks

In the conclusion, lets consider the following interesting

Example 4.1. In [2, Theorem 6.3] the following result is stated: Theorem 4.2. Denote X - is a moment-angle-manifold, corresponding to P (n, k ). Then X is dieomorphic to the connected sum of sphere products:
k j =1

j·

k+1 S j+1

2+j

вS

2n+k-j -1

.

In accordance with our main statement, the coecients in this formula give us an opportunity to nd the full Betti numbers for X, for which computation the following formula could be used:
dimk H k (ZP ) =
-i+2j =k



-i,2j

(P ).

From this fact we have:
b3 =
- 1,4

=

k+1 2

=

m-n , 2

which coincides with the result of the main statement with k = m - n - 1, i = 1, j = 2. If we nally analyse the numerical examples, considered above, we can proclaim the following:

Conjecture 4.3. For the n-dimensional Stashe polytope
the following take place:

-i,2(n+i-1)

K

n+2

, with n 2

= 0, i = 1, · · · , 2n - 5, = n + 3, if n - is even; n+3 if n - is odd. 2,

-(2n-4),2(3n-5)

It is possible that "minimal" properties of such type can be observed also in the cases of other nestohedra (stellohedra, permutohedra, cyclohedra), as the author has information in the case of n = 3, but he could not prove or even just formulate corresponding statements.

References
[1] V. M. Buchstaber,T. E. Panov.
Moscow, 2004. [2] Frederic Bosio and Laurent Meerssman.Real quadrics in convex polytopes. Acta Math. 197 (2006), no. 1, 53-127. [3] V. M. Buchstaber.Lectures on Toric Topology. Lecture notes of 'Toric Topology Workshop: KAIST 2008', in Trends in Mathematics. [4] Suyoung Choi and Jang Soo Kim.A combinatorial proof of a formula for betti numbers of a stacked polytope. arXiv:math.CO/0902.2444. Torus actions in topology and combinatorics.MCCME,

C

n

, complex manifolds and


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IVAN LIMONCHENKO
Department of Geometry and Topology, Faculty of Mathematics and Me-

chanics, Moscow State University, Leninskiye Gory, Moscow 119992, Russia
E-mail address

: iylim@mail.ru