Laplacians on cochains

Mostly standard. Purpose: fix formulas and inner-product conventions.

Definitions

Let $X$ be a finite simplicial complex. With the standard inner product on cochains, define the coboundary $\delta_k:C^k\to C^{k+1}$ and its adjoint $\delta_k^{\mathsf T}:C^{k+1}\to C^k$. Then $$L_k=\delta_{k-1}\delta_{k-1}^{\mathsf T}+\delta_k^{\mathsf T}\delta_k.$$

Identity

PSD and kernel

For all $x\in C^k$, $$\langle L_k x,x\rangle = \|\delta_{k-1}^{\mathsf T}x\|^2 + \|\delta_k x\|^2 \ \ge\ 0.$$ Hence $L_k\succeq 0$ and $$\ker L_k=\ker(\delta_{k-1}^{\mathsf T})\cap \ker(\delta_k).$$

Graph case ($k=0$)

For a graph $G=(V,E)$ with oriented edges, $\delta_0$ is the signed incidence matrix. Then $L_0=\delta_0^{\mathsf T}\delta_0$ is the usual graph Laplacian on vertices. For a $d$-regular graph, $L_0=dI-A$, so the “graph spectral gap” equals $d-\lambda_2(A)$.

Edge case ($k=1$) with triangles

If we attach a set of triangles $T$ to the base graph, then $$L_1(T)=\underbrace{\delta_0\delta_0^{\mathsf T}}_{L_1^{\downarrow}} +\underbrace{\delta_1(T)^{\mathsf T}\delta_1(T)}_{L_1^{\uparrow}(T)}.$$

Think of $L_1^{\downarrow}$ as measuring “vertex-driven” inconsistency of an edge signal, and $L_1^{\uparrow}$ as measuring “triangle curl.” Both are quadratic forms on $C^1$.

Interpretation is heuristic but often useful in proofs and experiments.

Rayleigh quotients

For PSD $A$, define $$\lambda_{\min}^+(A)=\min_{x\in(\ker A)^\perp,\ \|x\|=1}\langle Ax,x\rangle.$$ This is the right quantity when kernels vary with $T$.