Infinite-Dimensional Topology: Prerequisites and Introduction

Author: J. van Mill
Publisher Elsevier (North-Holland)
Series: North-Holland Mathematical Library, 43
Year: 1989
Review: AMS review

Description

The first part of this book is a text for graduate courses in topology. In chapters 1 - 5, part of the basic material of plane topology, combinatorial topology, dimension theory and ANR theory is presented. For a student who will go on in geometric or algebraic topology this material is a prerequisite for later work. Chapter 6 is an introduction to infinite dimensional topology; it uses for the most part geometric methods, and gets to spectacular results fairly quickly. The second part of this book, chapters 7 & 8, is part of geometric topology and is meant for the more advanced mathematician interested in manifolds. The text is self-contained for readers with a modest knowledge of general topology and linear algebra; the necessary background material is collected in chapter 1, or developed as needed. One can look upon this book as a complete and self-contained proof of Torunczyk's Hilbert cube manifold characterization theorem: a compact ANR X is a manifold modeled on the Hilbert cube if and only if X satisfies the disjoint-cells property. In the process of proving this result several interesting and useful detours are made.

Contents

  1. Extension Theorems

    2. Topological Spaces. Linear Spaces. Function Spaces. The Michael Selection Theorem and Applications. AR's and ANR's. The Borsuk Homotopy Extension Theorem.
  1. Elementary Plane Topology.

    2. The Brouwer Fixed-Point Theorem and Applications. The Borsuk-Ulam Theorem. The Poincaré Theorem. The Jordan Curve Theorem.
  1. Elementary Combinatorial Techniques.

    2. Affine Notions. Simplexes. Triangulation. Simplexes in Rn. The Brouwer Fixed-Point Theorem. Topologizing a Simplical Complex.
  1. Elementary Dimension Theory.

    2. The Covering Dimension. Zero-Dimensional Spaces. Translation into Open Covers. The Imbedding Theorem. The Inductive Dimension Functions ind and Ind. Mappings into Spheres. Totally Disconnected Spaces. Various Kinds of Infinite-Dimensionality.
  1. Elementary ANR Theory.

    2. Theory.Some Properties of ANR's. A Characterization of ANR's and AR's. Hyperspaces and the AR-Property. Open Subspaces of ANR's. Characterization of Finite-Dimensional ANR's and AR's. Adjunction Spaces of Compact A(N)R's.
  1. An Introduction to Infinite-Dimensional Topology.

    2. Constructing New Homeomorphisms from Old. Z-Sets. The Estimated Homeomorphism Extension Theorem for Compacta in s. The Estimated HomeomorphismExtension Theorem. Absorbers. Hilbert Space is Homeomorphic to the Countable Infinite Product of Lines. Inverse Limits. Hilbert Cube Factors.
  1. Cell-Like Maps and Q-Manifolds.

    2. Cell-Like Maps and Fine Homotopy Equivalences. Z-Sets in ANR's. The Disjoint-Cells Property. Z-Sets in Q-Manifolds. Torunczyk's Approximation Theorem and Applications. Cell-Like Maps. The Characterization Theorem.
  1. Applications. Infinite Products.

    2. Keller's Theorem. Cone Characterization of the Hilbert Cube. The Curtis-Schori-West Hyperspace Theorem.
What Next?
Bibliography
Subject Index