Since the efficient computability of the multiscale simplicial flat norm depends on the total unimodularity of the boundary matrix, we study the conditions under which total unimodularity of boundary matrices can be guaranteed. 

## 2.4 Simplicial Complexes and Relative Torsion 

Dey, Hirani, and Krishnamoorthy [14] have given a simple characterization of the sim- plicial complex whose boundary matrix is totally unimodular. In short, if the simplicial complex does not have relative torsion then its boundary matrix is totally unimodular. We state this and other related results here for the sake of completeness, and refer the reader to the original paper [14] for details and proofs. The simplicial complex K in these results has dimension d + 1 or higher. Recall that a d -dimensional simplicial complex is pure if it consists of d -simplices and their faces, i.e., there are no lower dimensional simplices that are not part of some d -simplex in the complex. 

Theorem 2.4.1. [14, Theorem 5.2] The boundary matrix [ ∂$_{d}$$_{+1}$ ] of a finite simplicial complex K is totally unimodular if and only if H$_{d}$ ( L, L$_{0}$ ) is torsion-free for all pure subcomplexes L$_{0}$, L of K , with L$_{0}$ ⊂ L . 

These authors further describe situations in which the absence of relative torsion is guaranteed. The following two special cases describe simplicial complexes for which the boundary matrix is always totally unimodular. 

Theorem 2.4.2. [14, Theorem 4.1] The boundary matrix [ ∂$_{d}$$_{+1}$ ] of a finite simplicial complex triangulating a compact orientable ( d + 1) -dimensional manifold is totally uni- modular. 

Theorem 2.4.3. [14, Theorem 5.7] The boundary matrix [ ∂$_{d}$$_{+1}$ ] of a finite simplicial complex embedded in$_{R}$ d +1 is totally unimodular. 

