Retraction properties of the orbit spaces
by
Sergey A. Antonian
Moscow State University

Let G denote a Hausdorff group. By a G-A(N)E we mean a G-space which is a G-equivariant absolute (neighborhood) extensor for the class of all metric G-spaces.

Theorem. Let G be a compact group, H be its closed normal subgroup and suppose that all the orbits of a G-space X are metric. Then if X is a G-A(N)E the H-orbit space X / H is a G-A(N)E. In particular the G-orbit space X / G is an A(N)E.

Consider the case when G=GL(n) the full linear group. Let E be the n-dimensional Euclidean space. Denote by N(E) the G-space of all norms f : E --> R endowed with the compact-open topology and with the natural G-action: (gf)(x) = f(g^-1x); g in G, f in N(E), x in E. The Banach-Mazur compactum Q(n) is just the orbit space N(E) / G.

The following corollary gives an affirmative answer to the first part of the well-known Problem 899, from the book Open Problems in Topology (ed. by J. van Mill and G. Reed):

Corollary. The Banach-Mazur compacta Q(n), n >= 1 are AR-spaces.

Date received: February 9, 1997

Copyright © 1997 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 # caak-04.