Ends of Complexes

A systematic exposition of the theory and practice of ends of manifolds and CW complexes, not previously available.

Bruce Hughes (Author), Andrew Ranicki (Author)

9780521055192, Cambridge University Press

Paperback / softback, published 21 January 2008

380 pages
22.8 x 15.1 x 2.1 cm, 0.576 kg

'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

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.

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.

Subject Areas: Algebraic topology [PBPD]