Helly, Cayley, Hypertrees, and Weighted Enumeration III

This is the third and last part of the journey from a Helly type conjecture of Katchalski and Perles to a Cayley’s type formula for “hypertrees”.  (On second thought I decided to divide it into two devoting the second to probabilistic questions.) This part will include several diversions, open problems, and speculations.  

11. How to make it work - the matrix tree theorem

Our high dimensional extension to Cayley’s theorem reads:

where the sum is over all d-dimensional simplicial complexes K on n labelled vertices, with a complete (d-1)-dimensional skeleton, and which are Q-acyclic, namely all their (reduced) homology groups with rational coefficients vanish.  

Looking at the various proofs of Cayley’s formula (there are many many many beautiful proofs), the proof that I know to apply is the one based on the matrix tree theorem.

Consider the signed incidence matrix A’ between all (d+1)-subsets and all d-subsets of {1,2,…,n} that represents the boundary operator of simplicial homology. The rank of this matrix is , and just like in the ordinary matrix tree theorem you delete rows to be left with linearly independent rows. Here you delete all rows corresponding to sets containing ‘n’ and you are left with a matrix A. Now we compute the determinant of  directly, and compare the result to a computation based on the Cauchy-Binet Formula.

The eigenvalues of  are the eigenvalues of the d-th Laplacian of the complete d-dimensional simplicial complex with n vertices. It is easy to inspect what they are and the determinant of is indeed .

The many square determinants correspond to d-dimensional simplicial complexes K on our labelled set of vertices, which satisfy and . Now if K has non-vanishing d-th homology, the determinant is zero. If K is a Q-acyclic simplicial complex (i.e., its (reduced) homology groups with rational coefficients are trivial) then it has a non zero determinant. So far, it is like trees, but next comes a surprise. The contribution of K is the square of the number of elements in , the (d-1)th homology group of K. This torsion group is finite, but for d >1 it need not be trivial.

Emily Peters presents the matrix-tree theorem. From “The Sarong Theorem Archive” - an electronic archive of images of people proving theorems while wearing sarongs.

 

12. An even simpler use of Cauchy-Binet worth knowing

Consider the matrix A, whose columns are all +1 -1 vectors of length n. Computing , via Cauchy-Binet Formula (or by other easy methods) asserts that the expected value  for all   +1/-1 matrices behaves roughly like . This was observed by Turan and Szekeres who also found a formula for the sum of the fourth powers of all 0-1 n by n matrices. See a leter paper by Turan. (I am not aware of a similar formula for the fourth power of the size of the homology groups for hypertrees.) Much is known about the determinant and related properties of random 0-1 matrices and the analogy between torsion in the homology groups of random complexes and determinants of random matrices looks like a good analogy.

13. Torsion

One consequence of the formula compared to the total number of available simplicial complexes is that the torsion group is typically huge. (For d>1, the expected value of for Q-acyclic complexes counted by the formula, is asymptotically larger than ; I am not aware of explicit examples with such a huge torsion group.)

How should we think about torsion in homology? It seems that thinking about the size of the torsion as a behaving like the determinant of a random matrix, may give a good intuition for many cases.

14. Extending other proofs for Cayley’s theorem?     

Cayley’s counting trees theorem has many wonderful proofs.  Can any other proof extend to the case of Q-acyclic simplicial complexes? For example, one proof relies on the exponential theorem that relates the exponential generating functions for connected and general graphs with a certain property P. (Followed by the Lagrange inversion formula.) Is there an analog of the exponential formula when connectivity is replaced by higher homology? Is there any analog of Prüfer sequences? I am not aware of any other proof that works.

15. Weights to the rescue of other conjectures? 

Can we use subtle weights to save other promising but false enumerative conjectures? The farthest reaching fantasy in this direction

Helly’s Theorem, “Hypertrees”, and Strange Enumeration II: The Formula

In the first part of this post we discussed an appealing conjecture regaring an extension of Cayley’s counting trees formula. The number of d-dimensional “hypertrees” should somehow add up  to  . But it was not clear to us which complexes we want to count and how. This counting problem started from a Helly type conjecture proposed by Katchalski and Perles.
For d=2 n=6 the situation was confusing. We had 46608 complexes that were collapsible. Namely, for these complexes it is possible to delete all triangles one at a time by removing in each step a triangle T and an edge E which is contained only in T. Once all triangles are removed we are left with a spanning tree on our 6 vertices. (Five out of the 15 edges survive).  In addition, there were 12 simplicial complexes representing 6-vertex triangulations of the real projective plane. 
We will continue the discussion in this part, show how the conjecture can be saved and at what cost. We will also discuss the solution of the Perles-Katchalski conjecture -  a Helly’s type conjecture that we started with.   In the third part we will explain the proof and mention further related results and problems, discuss higher Laplacians and their spectrum, and mention a few related probabilistic problems.  
 
 

 We ended part one with the question “What can we do?”

8. How to make the conjecture work

With such a nice conjecture we should not take no for an answer. To make the conjecture work we need to count each of the twelve 6-vertex triangulations of the real projective plane, four times. Four is the square of the number of elements in . This is the difference in higher dimensions, a Q-acyclic complex need not be Z-acyclic. Homology groups can have non trivial torsion.  In our case can be a non trivial finite group.
Here is the theorem:   

where the sum is over all d-dimensional simplicial complexes K on n labelled vertices, with a complete (d-1)-dimensional skeleton, and which are Q-acyclic, namely all their (reduced) homology groups with rational coefficients vanish.  
Looking at the various proofs of Cayley’s formula (there are many many many beautiful proofs and more), which one (or more) would you expect to extend to the high dimensional case? We will answer this question in part III. Can you guess?

Helly’s Theorem, “Hypertrees”, and Strange Enumeration I

1. Helly’s theorem and Cayley’s formula

Helly’s theorem asserts: For a family of n convex sets in , n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common.

Cayley’s formula asserts: The number of  trees on n labelled vertices is .

In this post (in two parts) we will see how an extension of Helly’s theorem has led to high dimensional analogs of Cayley’s theorem. 

  

left: Helly’s theorem demonstrated in the Stanford Encyclopedia of Philosophy (!), right: a tree 

2. Background

This post is based on my lecture at Marburg. The conference there was a celebration of new doctoral theses

Five Open Problems Regarding Convex Polytopes

  

The problems 

1. The conjecture

A centrally symmetric d-polytope has at least non empty faces.

2. The cube-simplex conjecture

For every k there is f(k) so that every d-polytope with has a k-dimensional face which is either a simplex or combinatorially isomorphic to a k-dimsnional cube.

3. Barany’s question

For every d-dimensional polytope P and every k, ,  is it true that ?

(In words: the number of k-dimensional faces of P is at least the minimum between the number of vertices of P and the number of facets of P. )

4.  Fat 4-polytopes

For 4-polytopes P, is the quantity  bounded from above by some absolute constant? 

5.  five-simplicial five-simple polytopes

Are there 5-simplicial 5-simple 10-polytopes? Or at least 5-simplicial 5-simple d-polytope for some d?

(A polytope P is k-simplicial if all its faces of dimension at most k, are simplices. A polytope P is k-simple if its dual P* is k-simplicial.)

And on a personal note: My beloved, beautiful,  and troubled country celebrates 60 today: happy birthday! 

Update (May 12): David Eppstein raised in a followup a sort of a dual question to Barany’s. For a d-polytope with n vertices and n facets what is the maximal number of k-faces. For a fixed d and large n the free join of pentagons is conjectured to give asymptotically the best upper bound.