Discrete Vector Fields and Fundamental Algebraic Topology

TL;DR

This paper demonstrates how key homology equivalences in fundamental Algebraic Topology can be constructively derived using discrete vector fields, providing explicit reductions where traditionally only non-constructive proofs exist.

Contribution

It introduces a systematic method to obtain constructive homology equivalences via discrete vector fields, enhancing the practical applicability of spectral sequence results.

Findings

Constructive derivation of homology equivalences

Application of discrete vector fields to spectral sequences

Explicit reductions in algebraic topology

Abstract

We show in this text how the most important homology equivalences of fundamental Algebraic Topology can be obtained as reductions associated to discrete vector fields. Mainly the homology equivalences whose existence -- most often non-constructive -- is proved by the main spectral sequences, the Serre and Eilenberg-Moore spectral sequences. On the contrary, the constructive existence is here systematically looked for and obtained.