Topological fundamental groupoid. I

TL;DR

This paper establishes a natural topology on the fundamental groupoid of certain spaces, showing it forms a topological groupoid, and explores its properties, including its relation to topological groups and local compactness.

Contribution

It introduces a natural topology on the fundamental groupoid for locally path connected semilocally simply connected spaces and clarifies its structural properties.

Findings

Fundamental groupoid can be equipped with a natural topology making it a topological groupoid.

The fundamental groupoid of a topological group is a transformation groupoid.

Fundamental groupoid is not etale, contrary to some beliefs.

Abstract

We show that the fundamental groupoid~\(\Pi_1(X)\) of a locally path connected semilocally simply connected space~\(X\) can be equipped with a \emph{natural} topology so that it becomes a topological groupoid; we also justify the necessity and minimality of these two hypotheses on~\(X\) in order to topologise the fundamental groupoid. We find that contrary to a belief -- especially among the Operator Algebraists -- the fundamental groupoid is not {\etale}. Further, we prove that the fundamental groupoid of a topological group, in particular a Lie group, is a \emph{transformation groupoid}; again, this result disproves a standard belief that the fundamental groupoids are \emph{far} away from being transformation groupoids. We also discuss the point-set topology on the fundamental groupoid with the intention of making it a locally compact groupoid.