Approximation by Equivariant Homeomorphisms. I

Abstract

Locally linear (= locally smoothable) actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other. For these actions, an equivariant $\alpha$-approximation theorem is proved using engulfing techniques. As corollaries are obtained equivariant "fibrations are bundles" and "controlled $h$-cobordism" theorems, as well as an equivariant version of Edwards’ cell-like mapping theorem and the vanishing of the set of transfer-invariant $G$-homotopy topological structures, rel boundary, on ${T^n} \times {D_\rho }$ (when ${T^n}$ is the $n$-torus with trivial $G$ action and ${D_\rho }$ is a representation disc).