Dense products in fundamental groupoids

TL;DR

This paper investigates the conditions under which dense products in fundamental groupoids are well-defined, revealing a distinction from fundamental groups and linking the property to the existence of generalized universal coverings.

Contribution

It demonstrates that well-defined dense products in fundamental groupoids are equivalent to the space admitting a generalized universal covering, unlike in fundamental groups.

Findings

Dense products in fundamental groupoids are not necessarily well-defined even if they are in fundamental groups.

The fundamental groupoid has well-defined dense products if and only if the space admits a generalized universal covering.

The paper clarifies the relationship between dense products and covering space properties in algebraic topology.

Abstract

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental group need not imply the well-definedness of countable dense products in the fundamental groupoid. Additionally, we show the fundamental groupoid $\Pi_1(X)$ has well-defined dense products if and only if $X$ admits a generalized universal covering space.