The first five chapters of this book form an introductory course in piece wise-linear topology in which no assumptions are made other than basic topological notions. This course would be suitable as a second course in topology with a geometric flavour, to follow a first course in point-set topology, andi)erhaps to be given as a final year undergraduate course. The whole book gives an account of handle theory in a piecewise linear setting and could be the basis of a first year postgraduate lecture or reading course. Some results from algebraic topology are needed for handle theory and these are collected in an appendix. In a second appen dix are listed the properties of Whitehead torsion which are used in the s-cobordism theorem. These appendices should enable a reader with only basic knowledge to complete the book. The book is also intended to form an introduction to modern geo metric topology as a research subject, a bibliography of research papers being included. We have omittedacknowledgements and references from the main text and have collected these in a set of "historical notes" to be found after the appendices. 1. Polyhedra and P.L. Maps.- Basic Notation.- Joins and Cones.- Polyhedra.- Piecewise-Linear Maps.- The Standard Mistake.- P. L. Embeddings.- Manifolds.- Balls and Spheres.- The Poincaré Conjecture and the h-Cobordism Theorem..- 2. Complexes.- Simplexes.- Cells.- Cell Complexes.- Subdivisions.- Simplicial Complexes.- Simplicial Maps.- Triangulations.- Subdividing Diagrams of Maps.- Derived Subdivisions.- Abstract Isomorphism of Cell Complexes.- Pseudo-Radial Projection.- External Joins.- Collars.- Appendix to Chapter 2. On Convex Cells.- 3. Regular Neighbourhoods.- Full Subcomplexes.- Derived Neighbourhoods.- Regular Neighbourhoods.- Regular Neighbourhoods in Manifolds.- Isotopy Uniqueness of Regular Neighbourhoods.- Collapsing.- Remarks on Simple Homotopy Type.- Shelling.- Orientation.- Connected Sums.- Schönflies Conjecture.- 4. Pairs of Polyhedra and Isotopies.- Links and Stars.- Collars.- Regular Neighbourhoods.- Simplicial Neighbourhood Theorem for Pairs.- Collapsing and Shellingfor Pairs.- Application to Cellular Moves.- Disc Theorem for Pairs.- Isotopy Extension.- 5. General Position and Applications.- General Position.- Embedding and Unknotting.- Piping.- Whitney Lemma and Unlinking Spheres.- Non-Simply-Connected Whitney Lemma.- 6. Handle Theory.- Handles on a Cobordism.- Reordering Handles.- Handles of Adjacent Index.- Complementary Handles.- Adding Handles.- Handle Decompositions.- The CW Complex Associated with a Decomposition.- The Duality Theorems.- Simplifying Handle Decompositions.- Proof of the h-Cobordism Theorem.- The Relative Case.- The Non-Simply-Connected Case.- Constructing h-Cobordisms.- 7. Applications.- Unknotting Balls and Spheres in Codimension ? 3.- A Criterion for Unknotting in Codimension 2.- Weak 5-Dimensional Theorems.- Engulfing.- Embedding Manifolds.- Appendix A. Algebraic Results.- A. 1 Homology.- A. 2 Geometric Interpretation of Homology.- A. 3 Homology Groups of Spheres.- A. 4 Cohomology.- A. 5 Coefficients.- A. 6 Homotopy Groups.- A. 8 The Universal Cover.- Appendix B. Torsion.- B. 1 Geometrical Definition of Torsion.- B. 2 Geometrical Properties of Torsion.- B. 3 Algebraic Definition of Torsion.- B. 4 Torsion and Polyhedra.- B. 5 Torsion and Homotopy Equivalences.- Historical Notes.
