The fundamental group as a topological group

TL;DR

This paper introduces a new topology on the fundamental group that makes it a topological group, capturing local space properties and aligning with classical algebraic topology results.

Contribution

It constructs a topology on the fundamental group using free topological groups, making it a topological group and enhancing its distinguishing power.

Findings

The invariant $mma_{1}^{\tau}$ is a topological group for any space.

It can distinguish spaces with isomorphic fundamental groups.

It aligns with the quotient topology when it forms a topological group.

Abstract

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted $\pi_{1}^{\tau}$, takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology.