This post is authored by Eran Nevo. (It is the first in a series of five posts.)
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 vertices and edges, where .
The 2-spheres with vertices have edges and triangles, and can be any integer . 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 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.
A collection of subsets of is called a (finite abstract) simplicial complex if it is closed under inclusion, i.e. implies . Note that if is not empty (which we will assume from now on) then , and is of dimension . The elements of are called faces. The -vector (face vector) of is where . For example, if is the -simplex then .
From now on, let’s assume that is a simplicial -sphere, meaning that its geometric realization is homeomorphic to the -dimensional sphere. What are the relations among the ‘s? We know that the reduced Euler characteristic is . There are more linear relations, called the Dehn-Sommerville relations. They have a nicer form when expressed in terms of the -vector of , defined by
We see that there are invertible maps .
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 5 12
1 4 7 8
h= (1 3 3 1)
Here, we start with the -vector of the Octahedron (1,6,12,8) (bold face entries) and take differences as shown in this picture to end with the -vector (1,3,3,1).
We see that . More is true. The Dehn-Sommerville relations state that is symmetric, i.e. for every . This result can be proved combinatorially, for the larger family of Eulerian posets, and actually these relations span all the linear equalities among the -vector of .
What about inequalities?
We say that a vector is an -vector ( for Macaulay) if it is the -vector of a multicomples, i.e. of a collection of multisets (elements can repeat!) closed under inclusion. For example, is an -vector, as is demonstrated by the multicomplex , written in monomial notation – the exponent tells how many copies of 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.
-vector and the -theorem
Let , for . . By the Dehn-Sommerville relations, can be recovered from . McMullen asked whether is always an -vector, and conjectured that this the case if is the boundary of a simplicial polytope. He conjectured further that any -vector is the -vector of the boundary of a simplicial polytope.
A major result is the proof of this conjecture in the polytopal case, known as the -theorem, which gives a complete characterization of the -vectors of boundaries of simplicial polytopes. Billera and Lee constructed in 1979 a simplicial polytope with boundary satisfying for any given -vector . Stanley proved, in the same year, that if is the boundary of a convex polytope then is an -vector. We will discuss this proof in the next section.
The -conjecture (or, one version of it) is:
If is a simplicial sphere then is an -vector.
This conjecture is around since McMullen first asked it in 1970.
Next time: how does Stanley’s proof go?