The Generalized Rank Invariant: M\"obius invertibility, Discriminating Power, and Connection to Other Invariants

In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of persistence for multi-parameter persistent homology is the rank invariant, which has recently evolved into the generalized rank invariant (GRI) by naturally extending its domain. This extension enables us to quantify persistence across a broader range of regions in the indexing poset compared to the rank invariant. However, the size of the domain of the GRI is generally formidable, making it desirable to restrict its domain to a more manageable subset for computational purposes. The foremost questions regarding such a restriction of the domain are: (1) How to restrict, if possible, the domain of the GRI without any loss of information? (2) When can we more compactly encode the GRI as a `persistence diagram'? (3) What is the trade-off between computational efficiency and the discriminating power of the GRI as the amount of the restriction on the domain varies? (4) What proxies exist for persistence diagrams in the multi-parameter setting that can be derived from the GRI? To address the first three questions, we generalize and axiomatize the classic fundamental lemma of persistent homology via the notion of M\"obius invertibility of the GRI which we propose. This extension also contextualizes known results regarding the (generalized) rank invariant within the classical theory of M\"obius inversion. We conduct a comprehensive comparison between M\"obius invertibility and other existing concepts related to the structural simplicity of persistence modules. We address the fourth question through the notion of motivic invariants. We demonstrate that many invariants from the literature can be both derived from the GRI and recast as motivic invariants.