The orientable pseudomanifold characterization via the top-dimensional Laplacian

Let $K$ be a $d$-dimensional simplicial complex. Write $K_d$ and $K_{d-1}$ for its sets of $d$-cells and $(d-1)$-cells, respectively, and let $\Delta_d$ be the $d$-dimensional Laplacian. Set

$$m=|K_d|,\qquad n=|K_{d-1}|.$$

Orientable pseudomanifold characterization. $K$ is an orientable pseudomanifold if and only if

$$\det(\Delta_d+\lambda I)=\sum_{i=0}^{n}|\mathcal{M}_{m-i}(K)|\lambda^{n-i}.$$

The conjecture proposes that the determinant identity characterizes orientable pseudomanifolds among all simplicial complexes. It is motivated by the observed particularly good behavior of discrete gradients on triangulated manifolds and simplicial pseudomanifolds; the paper gives evidence for the claim and announces future work addressing the connection in more detail.