Computing higher Leray-Serre spectral sequences of towers of fibrations

TL;DR

This paper introduces algorithms and a software implementation for computing higher Leray-Serre spectral sequences of towers of fibrations using effective homology, advancing computational tools in algebraic topology.

Contribution

It develops a novel algorithmic approach and implements it in Kenzo to compute these spectral sequences, extending computational capabilities for complex topological structures.

Findings

Successfully translated the spectral sequence construction into a simplicial framework

Implemented a new module in Kenzo for these computations

Demonstrated the effectiveness of the algorithms on relevant examples

Abstract

The higher Leray-Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray-Serre spectral sequence of a fibration. 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.