Retraction Properties of the Orbit Space
by
Sergey Antonian

Throughout G will denote a compact Hausdorff group. By a G-A(N)E (resp., G-A(N)E(k) where k >= 0 is an integer)-space we mean a G-space (not necessarily metric), which is a G-equivariant absolute (neighborhood) extensor for the class of all metric G-spaces M (resp. with dimM/G <= k).

Theorem Let N < G be a closed normal subgroup and suppose that all the orbits of a G-space X are metric. Then

1. if X is a G-A(N)E-space, then the N-orbit space X/N is a G-A(N)E-space. In particular, the G-orbit space X/G is an A(N)E-space,
2. if X is a G-A(N)E(k)-space, k >= 0, then X/N is a G-A(N)E(k)-space. In particular, X/G is a A(N)E(k)-space.

Corollary Let H be a subgroup of the symmetric group S_n, n >= 1. Then the functor SP_Hⁿ of symmetric n-th power associated with H preserves the properties of a G-space to be a G-A(N)E (resp. G-A(N)E(k), k >= 0)-space. In particular, SP_Hⁿ preserves the properties of a topological space to be an A(N)E (resp. A(N)E(k), k >= 0)-space.

Theorem Let G be a compact Lie group and 2^G be its hyperspace of closed subsets with the Hausdorff metric. Let G act on 2^G by left translations and let X = 2^G \{G}. Then the orbit space X/G is a Q-manifold.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-05.