I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the theory is that some spanning trees have torsion, and I want to highlight this fact. Here is the definition:

If $\Delta$ is a simplicial complex and $\Gamma^d$ is a subcomplex which agrees with $\Gamma$ in positive codimension $\Delta_{(d-1)}=\Gamma_{(d-1)}$, then $\Gamma$ is called a

spanning treeif any two (and hence all) of the following conditions hold:

- $\tilde H_d(\Gamma,\Bbb Z)=0$
- $|\tilde H_{d-1}(\Gamma,\Bbb Z)|<\infty$
- $f_d(\Gamma)=f_d(\Delta)-\tilde\beta_d(\Delta)+\tilde\beta_{d-1}(\Delta)$.

Ideally, I'm trying to find a simplicial complex which has both torsion and nontorsion spanning trees, and yet has few enough spanning trees that it could be reasonably computed on the board. Some candidates which don't fulfill these purposes include

- A minimal triangulation $\Bbb{RP}^2$ is a torsion tree, which shows existence but is not so interesting.
- You can stick a sphere out of $\Bbb{RP}^2$ (like the connect sum but without removing the disk) and this complex has four trees, but alas they are all torsion.
- There is a minimal triangulation of $\Bbb{RP}^2$ which spans the $2$-skeleton of the $6$-simplex, but there are far too many spanning trees of this complex to do on the board.