Fiberwise retraction and shape properties of the orbit space

From the point of view of retracts and shape theory, the category G-TOPB of G-spaces over a G-space B, where G is a compact group, is investigated. In particular, we prove that if B has only one orbit type and E is a metric G-ANR over B, then the orbit space E/G is an ANR over B/G. As an application we construct a fiberwise G-orbit functor μ : G-SHB → SHB/G on shape level.