Proper actions on topological groups: Applications to quotient spaces

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F of X such that FG=X and the restriction of the quotient projection X -> X/G to F is a perfect map F -> X/G. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from X to ferred from X/G to X. Yet another application leads to the inequality dim X<= dim X/G + dim G for every paracompact group X and its locally compact subgroup G.

💡 Research Summary

The paper investigates the natural left‑translation action of a locally compact subgroup G on a Hausdorff topological group X and shows that this action is proper in the sense of Palais. Properness means that for every compact subset K⊂X the set { g∈G | gK∩K≠∅ } is compact. The authors exploit the continuity of the group multiplication in X and the existence of a compact symmetric neighbourhood of the identity in G to verify this condition. Consequently, the action enjoys all the standard consequences of Palais‑proper actions, most notably the existence of slices (local cross‑sections) and the possibility of constructing a global closed transversal.

Using the slice theory, the authors construct a closed subset F⊂X such that every element of X can be written uniquely as fg with f∈F and g∈G; in symbols, X=F·G. The quotient projection π:X→X/G, when restricted to F, becomes a perfect map (continuous, closed, and with compact fibres). This perfectness is crucial because perfect maps preserve many important topological properties: if the quotient space X/G is paracompact, normal, or perfectly normal, then so is F, and hence X=F·G inherits the same property. Conversely, if X is paracompact, the openness of π implies that X/G is also paracompact. Thus a wide class of properties transfers back and forth between X and its quotient by G.

The second major application concerns covering dimension. For a paracompact group X and a locally compact subgroup G, the decomposition X=F·G together with dim F=dim X/G yields the inequality

 dim X ≤ dim X/G + dim G.

This inequality generalises the classical dimension formula for products of spaces and provides a quantitative measure of how the presence of a locally compact subgroup influences the overall dimension of the ambient group.

The paper also discusses concrete examples, such as ℝⁿ with closed subgroups, Lie groups with compact subgroups, and non‑abelian groups where the same machinery applies. These examples illustrate that the properness of the action is not limited to abelian or finite‑dimensional cases; it holds for any Hausdorff topological group with a locally compact subgroup.

In summary, the authors establish three interconnected results: (1) the left‑translation action of a locally compact subgroup on a Hausdorff topological group is Palais‑proper; (2) there exists a closed transversal F making the quotient projection a perfect map, which in turn enables the transfer of paracompactness, normality, and related properties between X and X/G; (3) the covering dimension of X is bounded above by the sum of the dimensions of the quotient and the subgroup. These findings deepen the relationship between group actions, quotient topology, and dimension theory, and they provide a robust framework for further investigations into more general group actions and non‑locally‑compact settings.