Ends of Complexes
Ends of Complexes
The ends of a topological space are the directions in which it becomes non-compact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behaviour at infinity of a non-compact space. The second part studies tame ends in topology. Tame ends are shown to have a uniform structure, with a periodic shift map. Approximate fibrations are used to prove that tame manifold ends are the infinite cyclic covers of compact manifolds. The third part translates these topological considerations into an appropriate algebraic context, relating tameness to homological properties and algebraic K- and L-theory.
- Ranicki is a well-known author
- This book ties up all the loose ends of the subject
- Features a blend of geometric topology, algebraic topology and algebra, designed to appeal to readers with background in only one of these disciplines
Reviews & endorsements
'The book gathers together the main strands of the theory of ends of manifolds from the last thirty years and presents a unified and coherent treatment of them. It also contains authoritative expositions of certain topics in topology such as mapping tori and telescopes, often omitted from textbooks. It is thus simultaneously a research monograph and a useful reference.' Proceedings of the Edinburgh Mathematical Society
'This is a highly specialized monograph which is very clearly written and made as accessible for the reader as possible … It is absolutely indispensable for any specialist in the field.' European Mathematical Society
Product details
No date availablePaperback
9780521055192
380 pages
228 × 151 × 21 mm
0.576kg
Table of Contents
- Introduction
- Chapter summaries
- Part I. Topology at Infinity:
- 1. End spaces
- 2. Limits
- 3. Homology at infinity
- 4. Cellular homology
- 5. Homology of covers
- 6. Projective class and torsion
- 7. Forward tameness
- 8. Reverse tameness
- 9. Homotopy at infinity
- 10. Projective class at infinity
- 11. Infinite torsion
- 12. Forward tameness is a homotopy pushout
- Part II. Topology Over the Real Line:
- 13. Infinite cyclic covers
- 14. The mapping torus
- 15. Geometric ribbons and bands
- 16. Approximate fibrations
- 17. Geometric wrapping up
- 18. Geometric relaxation
- 19. Homotopy theoretic twist glueing
- 20. Homotopy theoretic wrapping up and relaxation
- Part III. The Algebraic Theory:
- 21. Polynomial extensions
- 22. Algebraic bands
- 23. Algebraic tameness
- 24. Relaxation techniques
- 25. Algebraic ribbons
- 26. Algebraic twist glueing
- 27. Wrapping up in algebraic K- and L-theory
- Part IV. Appendices
- References
- Index.
