PROPERTIES OF Q AND OF Q-MANIFOLDS.

We let Q denote the Hilbert Qube: the countable infinite product of closed intervals. A Q-manifold is a separable metric space with an open cover of sets homeomorphic to open subsets of Q. Since Q is compact, every Q-manifold is locally compact. It is also an ANR.

Topological properties of Q and/or of Q-manifolds have been studied extensively and intensively over the past 10 or 12 years by T.A. Chapman, James E. West, Raymond Y-T Wong, L. Siebenmann, R.D. Edwards, S. Ferry, Z. Cerin, R.M. Schori, the author and several others. Introductory work on compact group actions on Q and on Q-manifolds have resulted in many interesting examples and one significant partial result by Wong. It is the purpose of this paper to survey the known results and techniques, to pose several open problems and to propose various areas for investigation.

The reader is referred to the book by Bessager and Pelcznski on Selected Topics in Infinite-Dimensional Topology (published, 1975, in Warsaw) and the CBMS Expository Lecture Notes on Q-Manifolds by T.A. Chapman (to be published in 1976 by the AMS) for background information and problems in infinite-dimensional topology.

We review below some of the properties of Q and of Q-manifolds used in posing problems and developing intuition about Q. In general, we do not give specific references but usually do attribute authorship to others by name.