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? Continue reading