## A Beautiful Garden of Hypertrees

We had a series of posts (1,2,3,4) “from Helly to Cayley” on weighted enumeration of Q-acyclic simplicial complexes. The simplest case beyond  Cayley’s theorem were Q-acyclic complexes  with $n$ vertices, ${n \choose 2}$ edges, and ${{n-1} \choose {2}}$ triangles. One example is the six-vertex triangulation of the real projective plane. But here, as in many other places, we are short of examples.

Nati Linial,  Roy Meshulam and Mishael Rosenthal wrote a paper with very surprising examples of Q-acyclic simplicial complexes called “sum complexes”. The basic idea is very simple: The vertices are $\{1,2,\dots , n\}$. Next you pick three numbers $a,b$ and $c$ and consider all the triples $i,j,k$ so that $i+j+k$ is either $a$ or $b$ or $c$. And let’s assume that $n$ is a prime.

So how many triangles do we have? A fraction of $3/n$ of all possible triangles which is what we want (${{n-1} \choose {2}}$).

If the three numbers form an arithmetic progression then the resulting simplicial complex is collapsible. In all other cases it is not collapsible. The proof that it is Q-acyclic uses a result of Chebotarëv on Fourier analysis. (So what does Fourier analysis have to do with computing homology? You will have to read the paper!) The paper considers the situation in all dimensions.

What about such combinatorial constructions for Q-homology spheres?

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