On the Topology of J-Groups

TL;DR

This paper introduces the concept of topological J-groups, explores their properties, and shows that pathwise connected topological J-groups are weakly contractible, revealing a significant topological restriction.

Contribution

It defines topological J-groups and characterizes their topological and homotopical properties, including the weak contractibility of pathwise connected examples.

Findings

Many important topological groups are not topological J-groups

Pathwise connected topological J-groups are weakly contractible

Weak contractibility depends solely on the homotopy type

Abstract

We introduce the concept of a topological J-group and determine for many important examples of topological groups if they are topological J-groups or not. Besides other results, we show that the underlying topological space of a pathwise connected topological J-group is weakly contractible which is a strong and unexpected obstruction that depends only on the homotopy type of the space.