"Small" simplicial complex with torsion trees

Question

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 tree if 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.

Answer 1

Here is an example (which might be be what you are describing in your third bullet point) that I think is both doable and illustrates torsion spanning trees. Consider the minimal triangular of $\mathbb{RP}^2$ shown below.

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Let $\Gamma$ denote this simplicial complex which triangulates $\mathbb{RP}^2$. Let $\Delta = \Gamma \cup \{1,2,3\}$ and let $\Theta = \Delta \setminus \{4,5,6\}$. Then $\Gamma$ is a simplicial spanning tree of $\Delta$ with torsion while $\Theta$ is a simplicial spanning tree of $\Delta$ with out torsion. Further $\Theta' = \Delta \setminus \{i,j,k\}$ for any $\{i,j,k\} \ne \{1,2,3\}$ will be homotopy equivalent to $\Theta$ and thus have the same homology.