(Eran Nevo) The g-Conjecture I

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

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.

A good place to read more about it is Stanley’s book Combinatorics and Commutative algebra‘. First let’s fix some notation.

$f \leftrightarrow h$ vectors

A collection $K$ of subsets of $[n]=\{1,2,...,n\}$ is called a (finite abstract) simplicial complex if it is closed under inclusion, i.e. $S\subseteq T\in K$ implies $S\in K$. Note that if $K$ is not empty (which we will assume from now on) then $\emptyset \in K$, and is of dimension $-1$. The elements of $K$ are called faces. The $f$vector (face vector) of $K$ is $f(K)=(f_{-1},f_0,f_1,...)$ where $f_{i-1}=|\{T\in K: |T|=i\}|$. For example, if $K$ is the $(d-1)$-simplex then $f_{i-1}(K)=\binom{d}{i}$.

From now on, let’s assume that $K$ is a simplicial $(d-1)$-sphere, meaning that its geometric realization is homeomorphic to the $(d-1)$-dimensional sphere. What are the relations among the $f_j(K)$‘s? We know that the reduced Euler characteristic is $\tilde{\chi}(K)=\sum_{-1\leq i\leq d-1}(-1)^i f_{i}(K)=(-1)^{d-1}$. There are more linear relations, called the Dehn-Sommerville relations. They have a nicer form when expressed in terms of the $h$vector of $K$, defined by

$\sum_{0\leq i\leq d}h_i(K)x^{d-i}= \sum_{0\leq i\leq d}f_{i-1}(K)(x-1)^{d-i}.$

We see that there are invertible maps $f(K) \leftrightarrow h(K)$.

What’s called “Stanley’s trick” is a convenient way to practically compute one from the other, as illustrated in the difference table below, taken from Ziegler’s book Lectures on Polytopes’, p.251:

1

1           6

1          5            12

1          4           7             8

h= (1        3          3            1)

Here, we start with the $f$-vector of the Octahedron (1,6,12,8) (bold face entries) and take differences as shown in this picture to end with the $h$-vector (1,3,3,1).

We see that $h_d(K)=(-1)^{d-1}\tilde{\chi}(K)=1=h_0(K)$. More is true. The Dehn-Sommerville relations state that $h(K)$ is symmetric, i.e. $h_i(K)=h_{d-i}(K)$ for every $0\leq i\leq d$. This result can be proved combinatorially, for the larger family of Eulerian posets, and actually these relations span all the linear equalities among the $f$-vector of $K$.

$M$-vectors

We say that a vector $h$ is an $M$-vector ($M$ for Macaulay) if it is the $f$-vector of a multicomples, i.e. of a collection of multisets (elements can repeat!) closed under inclusion. For example, $h=(1,1,1)$ is an $M$-vector, as is demonstrated by the multicomplex $\{1=x^0,x,x^2\}$, written in monomial notation – the exponent tells how many copies of $x$ to take. Macaulay gave a numerical characterization of such vectors. (The proof uses compression – see this post for a general description of the method.) We will revisit Macaulay theorem in the next part, when discussing face rings.

$g$-vector and the $g$-theorem

Let $g_0(K):=h_0(K)=1$, $g_i(K):=h_i(K)-h_{i-1}(K)$ for $1\leq i\leq \lfloor d/2\rfloor$. $g(K):=(g_0(K),...,g_{\lfloor d/2\rfloor}(K))$. By the Dehn-Sommerville relations, $f(K)$ can be recovered from $g(K)$. McMullen asked whether $g(K)$ is always an $M$-vector, and conjectured that this the case if $K$ is the boundary of a simplicial polytope. He conjectured further that any $M$-vector is the $g$-vector of the boundary of a simplicial polytope.

A major result is the proof of this conjecture in the polytopal case, known as the $g$-theorem, which gives a complete characterization of the $f$-vectors of boundaries of simplicial polytopes. Billera and Lee constructed in 1979 a simplicial polytope with boundary $K$ satisfying $g(K)=m$ for any given $M$-vector $m$. Stanley proved, in the same year, that if $K$ is the boundary of a convex polytope then $g(K)$ is an $M$-vector. We will discuss this proof in the next section.

The $g$-conjecture (or, one version of it) is:

If $K$ is a simplicial sphere then $g(K)$ is an $M$-vector.

This conjecture is around since McMullen first asked it in 1970.

Next time: how does Stanley’s proof go?