Singular chains and the fundamental group

TL;DR

This paper demonstrates that the algebraic structure of singular chains on a path connected space uniquely determines its fundamental group and introduces a notion of weak equivalence that preserves this data.

Contribution

It establishes a functorial relationship between singular chains and the fundamental group, and defines weak equivalences maintaining this structure.

Findings

Singular chains' algebraic structure determines the fundamental group.

A notion of weak equivalence preserves the fundamental group data.

The fundamental group can be recovered functorially from singular chains.

Abstract

We show that the natural algebraic structure of the singular chains on a path connected topological space determines the fundamental group functorially. Moreover, we describe a notion of weak equivalence for the relevant algebraic structure under which the data of the fundamental group is preserved.