The fundamental group as a topological group

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.

💡 Research Summary

The paper addresses a long‑standing deficiency in classical algebraic topology: the fundamental group $\pi_{1}(X)$, when equipped with the quotient topology inherited from the loop space $\Omega X$, often fails to be a topological group because the multiplication map is not continuous. To remedy this, the author introduces a canonical construction that endows the fundamental group of any space with a genuine topological‑group structure, denoted $\pi_{1}^{\tau}(X)$.

The key idea is to replace the naïve quotient by a construction based on free topological groups. For a space $X$, the loop space $\Omega X$ carries the compact‑open topology. The free topological group $F_{\mathrm{top}}(\Omega X)$ is the universal topological group generated by $\Omega X$: it is obtained by first forming the algebraic free group on the set $\Omega X$ and then imposing the weakest topology that makes the inclusion $\Omega X\hookrightarrow F_{\mathrm{top}}(\Omega X)$ continuous and the group operations continuous. This universal property ensures that any continuous map from $\Omega X$ into a topological group factors uniquely through $F_{\mathrm{top}}(\Omega X)$.

Inside $F_{\mathrm{top}}(\Omega X)$ one considers the normal closed subgroup $N$ generated by all elements of the form $\alpha\beta^{-1}$ where $\alpha,\beta\in\Omega X$ are homotopic loops (i.e., represent the same element of the ordinary fundamental group). Because $N$ is closed, the quotient $F_{\mathrm{top}}(\Omega X)/N$ inherits a natural topological‑group structure. By definition
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