Computing invariants for multipersistence via spectral systems and effective homology (Q2229742)

This paper deals with new computations of multipersistence over posets. The main building blocks are the use of spectral systems, generalising spectral sequences to any \(I\)-filtration for \(I\) a poset, and the Kenzo software. Interestingly, the new programs of the paper are more general than existing implementations, since they allow computing multipersistence over integer coefficients with filtrations over any poset. The effective homology technique of Kenzo also allows to determine multipersistence for non-finitely generated filtered chain complexes. Recall that given an \(I\)-filtration \((F_i)_{i \in I}\) of a chain complex and \(v \leq w\) in \(I\), we can define \(\beta_n(v, w) = \text{dim} \ Im(H_n(F_v) \rightarrow H_n(F_w))\), where the map in homology is induced by the inclusion \(F_v \hookrightarrow F_w\). The rank invariant is the collection of all \(\beta_n(v, w)\), for any \(n\) and any \(v \leq w\). In the paper it is shown that the dimensions of the terms in the associated spectral system can be expressed as a combination of some \(\beta_n(v, w)\). The authors present a new invariant for multipersistence coming from the spectral system of the downset filtration, instead of a standard filtration by a poset. It is shown by example that this invariant allows to discriminate between a larger number of topological features in a filtered chain complex than the classical rank invariant. This paper should appeal to those working in algorithmic aspects of computing multipersistence. The paper shows that the reach of descriptors for multipersistence can be extended, as well as expanding to more complicated non-finitely generated spaces by using new software tools.