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.
Topological Spaces. Linear Spaces. Function Spaces. The Michael Selection
Theorem and Applications. AR's and ANR's. The Borsuk Homotopy Extension
Elementary Plane Topology.
The Brouwer Fixed-Point Theorem and Applications. The Borsuk-Ulam Theorem.
The Poincaré Theorem. The Jordan Curve Theorem.
Elementary Combinatorial Techniques.
Affine Notions. Simplexes. Triangulation. Simplexes in Rn. The Brouwer
Fixed-Point Theorem. Topologizing a Simplical Complex.
Elementary Dimension Theory.
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
Elementary ANR Theory.
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
An Introduction to Infinite-Dimensional Topology.
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.
Cell-Like Maps and Q-Manifolds.
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.
Applications. Infinite Products.
Keller's Theorem. Cone Characterization of the Hilbert Cube. The Curtis-Schori-West