Monodromy in bi-parameter persistence modules

Informally, monodromy captures the behavior of objects when one circles around a singularity. In persistent homology, non-trivial monodromy has been observed in the case of biparameter filtrations obtained by sublevel sets of a continuous function [1]. One might consider the fundamental group of an admissible open subspace of all lines defining linear one-parameter reductions of a bi-parameter filtration. Monodromy occurs when this fundamental group acts non-trivially on the persistence space, i.e. the collection of all the persistence diagrams obtained for each linear one-parameter reduction of the bi-parameter filtration. Here, under some tameness assumptions, we formalize the monodromy behavior in algebraic terms, that is in terms of the persistence module associated with a bi-parameter filtration. This allows to translate monodromy in terms of persistence module presentations as bigraded modules. We prove that non-trivial monodromy involves generators within the same summand in the direct sum decomposition of a persistence module. Hence, in particular interval-decomposable persistence modules have necessarily trivial monodromy group.