$G\operatorname{-CGWH}$ spaces and their (homotopy) colimits

TL;DR

This paper explores the properties of $G$-CGWH spaces with group actions, focusing on colimits, homotopy colimits, and their relationships with mapping spaces and homotopy groups.

Contribution

It provides new insights into the behavior of colimits and homotopy colimits in the context of $G$-CGWH spaces with Lie group actions.

Findings

Relations between (homotopy) colimits and mapping spaces established

Connections between homotopy colimits and homotopy groups clarified

Framework for analyzing $G$-spaces with colimits developed

Abstract

In this paper we take a look at compactly generated weak Hausdorff spaces equipped with an action of a compact Lie group $G$ together with their colimits and homotopy colimits. In particular, we investigate relations between (homotopy) colimits and mapping spaces, and consequently, homotopy groups.