# Lectures on Hilbert Cube Manifolds

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*T. A. Chapman*

A co-publication of the AMS and CBMS

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.

#### Table of Contents

# Table of Contents

## Lectures on Hilbert Cube Manifolds

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Introduction ix10 free
- I. Preliminaries 112 free
- II. Z-Sets in Q 718
- III. Stability of Q-Manifofds 1829
- IV. Z-Sets in Q-Manifolds 2536
- V. Q-Manifolds of the Form M x [0,1) 3243
- VI. Shapes of Z-Sets in Q 3950
- VII. Near Homeomorphisms and the Sum Theorem 4556
- VIII. Applications of the Sum Theorem 5465
- IX. The Splitting Theorem 6071
- X. The Handle Straightening Theorem 7081
- XI. The Triangulation Theorem 8192
- XII. The Classification Theorem 8697
- XIII. Cell-Like Mappings 91102
- XIV. The ANR Theorem 99110
- References 108119
- Appendix. Open Problems in Infinite-Dimensional Topology 111122
- Back Cover Back Cover1145

#### Readership

#### Reviews

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

-- Mathematical Reviews