Shape Morphisms to Transitive $G$-Spaces

The following problem plays an important role in shape theory: find conditions that guarantee that a shape morphism $F\colon X\mapsto Y$ of a topological space $X$ to a topological space $Y$ is generated by a continuous mapping $f\colon X\mapsto Y$. In the present paper, we study this problem in equivariant shape theory and give a solution for shape-equivariant morphisms to transitive $G$-spaces, where $G$ is a compact group with countable base. As a corollary, we prove a sufficient condition for equivariant shapes of a $G$-space $X$ to be equal to the group $G$ itself. We also prove some statements concerning equivariant bundles that play the key role in the proof of the main results and are of interest on their own.