Diffeomorphism of complements to compact sets in Euclidean spaces (Q748969)

Let G be a compact Lie group, \(\nu\) be an n-dimensional orthogonal representation of G and \(E^ n_{\nu}\) the resulting G-space which is assumed to have a nontrivial fixed point. Let \(E=(E^ n_{\nu})^{2k+r}\), \(r\geq 2\), with coordinatewise action of G. A compactum X in E is called \(k\nu\)-pseudo-smooth if it is G-invariant and if there exist smooth compact G-invariant submanifolds \(M_ i\) in E with dim \(M_ i\leq k\) and G-invariant tubular neighbourhoods \(N_ i\) of \(M_ i\) such that \(N_{i+1}\subset int N_ i\) and \(\cap^{\infty}_{i=1}N_ i=X\). The present paper contains the following theorem. If A, B are \(k\nu\)-pseudo-smooth compacta in E which have the same G-equivariant shape, then the complements E-A and E-B are G-equivariantly diffeomorphic. This is one half of an equivariant version of the well-known finite-dimensional Chapman complement theorem in shape theory.