(Eran Nevo) The g-Conjecture III: Algebraic Shifting

This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. Eran’s second post was about the commutative-algebra content of the conjecture. It described the Cohen-Macaulay property (which is largely understood and known to hold for simplicial spheres) and the Lefshetz property which is known for simplicial polytopes and is wide open for simplicial spheres.

The g-conjecture and algebraic shifting

Squeezed spheres

Back to the question from last time, Steinitz showed that

any simplicial 2-sphere is the boundary of a convex 3-polytope.

However, in higher dimension

there are many more simplicial spheres than simplicial polytopes,

on a fixed large number of vertices. Continue reading

(Eran Nevo) The g-Conjecture II: The Commutative Algebra Connection

Richard_Stanley-1

Richard Stanley

This post is authored by Eran Nevo. (It is the second in a series of five posts.)

The g-conjecture: the commutative algebra connection

Let K be a triangulation of a (d-1)-dimensional sphere. Stanley’s idea was to associate with K a ring R, and study the relations between algebraic properties of R and combinatorial properties of K.

Face ring

Fix a field k. The face ring (Stanley-Reisner ring) of K over k is k[K]=k[x_{1},..,x_{n}]/I_{K} where I_{K} is the homogenous ideal generated by the monomials whose support is not in K, \{\prod_{i\in S}x_i:\ S\notin K\}. For example, if K is the boundary of a triangle, then k[K]=k[x,y,z]/(xyz). k[K] is graded by degree (variables have degree one, 1 has degree zero), and let’s denote the degree i part by k[K]_i. This part is a finite dimensional k-vector space and we can collect all these dimensions in a sequence, or a series, called the Hilbert series of k[K], which carries the same information as f(K). More precisely,

hilb(k[K]):=\sum_{i\geq 0}\dim_k k[K]_i t^i = \frac{h_0(K)+h_1(K)t+...+h_d(K)t^d}{(1-t)^d}

(recall that K is (d-1)-dimensional).

Cohen-Macaulay (CM) ring

The ring k[K] is called Cohen Macaulay (CM) if there are d elements \Theta=\{\theta_1,..,\theta_d\} in k[K]_1 such that k[K] is a free k[\Theta]-module. As hilb(k[\Theta])=\frac{1}{(1-t)^d}, the numerical consequence is that hilb(k[K]/(\Theta))=h(K) (we use h both as a vector and as a polynomial, with the obvious identification).

Macaulay (revisited) showed that the Hilbert series of standard rings (=quatient of the polynomial ring by a homogenous ideal) are exactly the M-vectors (sequences).

A theorem of Riesner characterizes the simplicial complexes K with a CM face ring over a fixed field k in terms of the homology of K and its face links (with $k$-coefficients). It follows that if K is a simplicial sphere then k[K] is CM, hence h(K) is an $M$ vector! This gives more inequalities on f(K). This is also how Stanley proved the Upper Bound Conjecture, for face number of spheres: It follows that if K is a (d-1)-sphere with n vertices, and C(d,n) is the boundary of the cyclic d-polytope with n vertices, then for every i, f_i(K)\leq f_i(C(d,n)). This is as h_i(K)\leq \binom{n-d+i-1}{i}=h_i(C(d,n)).

Hard Lefschetz

Let K be the boundary of a simplicial d-polytope. Stanley observed that the hard Lefschetz theorem for toric varieties, an important theorem in algebraic geometry, translates in the language of face rings as follows: there exists \Theta as above and \omega\in k[K]_1 such that the maps

w^{d-2i}: (k[K]/(\Theta))_i\rightarrow (k[K]/(\Theta))_{d-i}

are isomorphisms between those vector spaces for any integer 0\leq i\leq \frac{d}{2}. In particular, w: (k[K]/(\Theta))_{i-1}\rightarrow (k[K]/(\Theta))_{i} is injective for 1\leq i\leq \frac{d}{2}. Thus, the quotient ring k[K]/(\Theta, \omega) has Hilbert series starting with g(K). This means, again by Macaulay theorem, that g(K) is an M-vector!

Later, in 1993, McMullen gave a different proof of this part of his conjectured g-theorem. His proof actually proves hard Lefschetz for this case. See McMullen’s survey paper `Polyhedra and polytopes: algebra and combinatorics’.

Problems

Does hard Lefschetz theorem hold for non polytopal spheres?

Can you think of examples of simplicial spheres which cannot be realized as the boundary of convex polytopes?

How the g-Conjecture Came About

This post complements Eran Nevo’s first  post on the g-conjecture

1) Euler’s theorem

euler1

Euler

Euler’s famous formula V-E+F=2 for the numbers of vertices, edges and faces of a  polytope in space is the starting point of many mathematical stories. (Descartes came close to this formula a century earlier.) The formula for d-dimensional polytopes P is

f_0(P)-f_1(P)+f_2(P)+\dots+(-1)^{d-1}f_{d-1}(P)=1-(-1)^d.

The first complete proof (in high dimensions) was provided by Poincare using algebraic topology. Earlier geometric proofs were based on “shellability” of polytopes which was only proved a century later. But there are elementary geometric proofs that avoid shellability.

2) The Dehn-Sommerville relations

Dehn

Consider a 3-dimensional simplicial polytope, – Continue reading

(Eran Nevo) The g-Conjecture I

This post is authored by Eran Nevo. (It is the first in a series of five posts.)

mcmullen

Peter McMullen

The g-conjecture

What are the possible face numbers of triangulations of spheres?

There is only one zero-dimensional sphere and it consists of 2 points.

The one-dimensional triangulations of spheres are simple cycles, having n vertices and n edges, where n\geq 3.

The 2-spheres with n vertices have 3n-6 edges and 2n-4 triangles, and n can be any integer \geq 4. This follows from Euler formula.

For higher-dimensional spheres the number of vertices doesn’t determine all the face numbers, and for spheres of dimension \geq 5 a characterization of the possible face numbers is only conjectured! This problem has interesting relations to other mathematical fields, as we shall see. Continue reading

Billerafest

Lou Billera
I am unable to attend the conference taking place now at Cornell, but I send my warmest greetings to Lou from Jerusalem. The titles and abstracts of the lectures can be found here. Let me tell you about two theorems by Lou.
 
The first is the famous g-theorem: The g-theorem is a complete description of f-vectors (= vecors of face numbers) of simplicial d-polytopes. This characterization was proposed by Peter McMullen in 1970, and it was settled in two works. Billera and Carl Lee proved the sufficiency part of McMullen’s conjecture, namely for every sequence of numbers which satisfies McMullen’c conjecture they constructed a simplicial d-polytope P whose f-vector is the given sequence. Richard Stanley proved the necessity part based on the hard Lefschetz theorem in algebraic geometry. The assertion of the g-conjecture (the necessity part) for triangulations of spheres is open, and this is probably the one single problem I spent the most time on trying to solve. 
 
The second theorem is a beautiful theorem by Margaret Bayer and Billera. Consider general d-polytopes. for a set S \subset {0,1,2,…,d-1}, S={ i_1,i_2,\dots,i_k} , i_1<i_2< \dots, define the flag number f_S as the number of chains of faces F_1 \subset F_2 \subset \dots F_k, where \dim F_j=i_j.  Bayer and Billera proved that the affine dimension of flag numbers of d-polytopes is c_d-1 where c_d is the dth Fibonacci number. (c_1=1, c_2=2, c_3=3, c_4=5, etc.) The harder part of this theorem was to construct c_d d-polytopes whose sequences of flag numbers are affinely independent.  The construction is simple: It is based on polytopes expressed by words of the form PBBPBPBBBPBP  where you start with a point, and P stands for “take a pyramid” and B stands for “take a bipyramid.” And the word starts with a P (to the left) and has no two consecutive B’s.
 
Let’s practice the notions of f-vectors and flag vectors on the 24-cell   . (The figure is a 3-dimensional projection into one of the facets of the polytope.)
 
 
 
This  4-polytope has 24 octahedral facets. It is self dual.
So f_0=f_3=24. And f_1=f_2=96. And f_{02}=288, f_{03}=192, etc.