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AuthorRourke, Colin P. author
TitleIntroduction to Piecewise-Linear Topology [electronic resource] / by Colin P. Rourke, Brian J. Sanderson
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1982
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Descript VIII, 126 p. online resource


The first five chapters of this book form an introductory course in pieceยญ 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 piecewiseยญ 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 appenยญ 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 geoยญ metric topology as a research subject, a bibliography of research papers being included. We have omitted acknowledgements 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 Poincarรฉ 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 -- Schรถnflies Conjecture -- 4. Pairs of Polyhedra and Isotopies -- Links and Stars -- Collars -- Regular Neighbourhoods -- Simplicial Neighbourhood Theorem for Pairs -- Collapsing and Shelling for 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

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