Computing higher Leray-Serre spectral sequences of towers of fibrations (Q2231647)

Spectral systems and higher spectral sequences are generalizations of spectral sequences due to [\textit{B.\ Matschke}, ``Successive spectral sequences'', Preprint, \url{arXiv:1308.3187}; ``Higher spectral sequences'', Preprint, \url{arXiv:2107.02130}] that arise from filtrations by arbitrary posets or multiple but compatible filtrations of a single chain complex. As this extra generality comes at the cost of extra bookkeeping, algorithms for computations with higher spectral sequences are a worthwhile target. In the article under review, the authors continue their investigation of algorithms for higher spectral sequences begun in [\textit{A. Guidolin} and \textit{A. Romero}, in: Proceedings of the 43rd international symposium on symbolic and algebraic computation, ISSAC 2018, New York, NY, USA, July 16--19, 2018. New York, NY: Association for Computing Machinery (ACM). 183--190 (2018; Zbl 1467.55011)] and devise algorithms for computing with Matschke's higher Leray-Serre spectral sequence associated with a tower of fibrations. From the abstract: ``In this work, we present algorithms to compute higher Leray-Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the higher Leray-Serre spectral sequence in a simplicial framework and studied some of its fundamental properties.''