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Lectures on Hilbert Cube Manifolds
T. A. Chapman, University of Kentucky, Lexington, KY
A co-publication of the AMS and CBMS.
 SEARCH THIS BOOK:
CBMS Regional Conference Series in Mathematics
1976; 131 pp; softcover
Number: 28
Reprint/Revision History:
reprinted 1982
ISBN-10: 0-8218-1678-0
ISBN-13: 978-0-8218-1678-3
List Price: US$25 Member Price: US$20
All Individuals: US\$20
Order Code: CBMS/28

The goal of these lectures is to present an introduction to the geometric topology of the Hilbert cube Q and separable metric manifolds modeled on Q, which are called here Hilbert cube manifolds or Q-manifolds. In the past ten years there has been a great deal of research on Q and Q-manifolds which is scattered throughout several papers in the literature. The author presents here a self-contained treatment of only a few of these results in the hope that it will stimulate further interest in this area. No new material is presented here and no attempt has been made to be complete. For example, the author has omitted the important theorem of Schori-West stating that the hyperspace of closed subsets of $$[0,1]$$ is homeomorphic to Q. In an appendix (prepared independently by R. D. Anderson, D. W. Curtis, R. Schori and G. Kozlowski) there is a list of problems which are of current interest. This includes problems on Q-manifolds as well as manifolds modeled on various linear spaces. The reader is referred to this for a much broader perspective of the field.

In the first four chapters, the basic tools which are needed in all of the remaining chapters are presented. Beyond this there seem to be at least two possible courses of action. The reader who is interested only in the triangulation and classification of Q-manifolds should read straight through (avoiding only Chapter VI). In particular the topological invariance of Whitehead torsion appears in Section 38. The reader who is interested in R. D. Edwards' recent proof that every ANR is a Q-manifold factor should read the first four chapters and then (with the single exception of 26.1) skip over to Chapters XIII and XIV.

Reviews

"This is an important contribution, since it compiles known results from a variety of papers into one well-written source."

-- Mathematical Reviews

• Preliminaries
• Z-sets in Q
• Stability of Q-manifoldsitle>
• Z-sets in Q-manifolds
• Q-manifolds of the form $$M \times [0, 1)$$
• Shapes of Z-sets in Q
• Near homeomorphisms and the Sum Theorem
• Applications of the Sum Theorem
• The Splitting Theorem
• The Handle Straightening Theorem
• The Triangulation Theorem
• The Classification Theorem
• Cell-like mappings
• The ANR Theorem
• References
• Appendix Open problems in infinite-dimensional topology