Let $\Sigma$ be an abstract simplicial complex. Do the eigenvectors and eigenvalues of the combinatorial Hodge Laplacian $\Delta_k$,
$$\Delta_k^\Sigma = (\partial_k^\Sigma)^\dagger \partial_k^\Sigma + \partial_{k+1}^\Sigma (\partial_{k+1}^\Sigma)^\dagger$$
have an interpretation in the field of algebraic topology? For example, the chains $f$ that are solutions to the Laplace equation $\Delta f = 0$ can be interpreted as the representative of the homology class and the shortest cycle around a certain $k$-hole. Can the other eigenchains, whose eigenvalue is $>0$, be interpreted in some way? Is an eigenchain with a corresponding eigenvalue slightly bigger than zero different from an instance with a large eigenvalue associated?
It seems to me that cohomology theory allows for an interesting interpretation, where $f \in \mathcal{C}^k$ cochain are eigenvectors of the combinatorial Laplacian, $\Delta f = \lambda f$, and $\lambda$ is interpreted as the frequency of the function (e.g., here). Can we derive an interpretation in algebraic topology from this theory?
A possible input here:
- Near-harmonic forms and quantitative topology: The operation of the Laplacian on a co-chain indicates how close it is to being a cohomology class. This lets one to quantify the fragilities of a simplicial complex. The eigenvectors corresponding to small non-zero eigenvalues indicate what parts of the complex are close to becoming holes. This is a more useful abstraction of network holes than merely knowing their absence or presence. Work needs to be done in order to quantify these notions more carefully