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Surgery on Compact Manifolds: Second Edition
C. T. C. Wall, University of Liverpool, England, and A. A. Ranicki (Editor), University of Edinburgh, Scotland
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Mathematical Surveys and Monographs
1999; 302 pp; hardcover
Volume: 69
ISBN-10: 0-8218-0942-3
ISBN-13: 978-0-8218-0942-6
List Price: US$72 Member Price: US$57.60
Order Code: SURV/69

The publication of this book in 1970 marked the culmination of a particularly exciting period in the history of the topology of manifolds. The world of high-dimensional manifolds had been opened up to the classification methods of algebraic topology by Thom's work in 1952 on transversality and cobordism, the signature theorem of Hirzebruch in 1954, and by the discovery of exotic spheres by Milnor in 1956.

In the 1960s, there had been an explosive growth of interest in the surgery method of understanding the homotopy types of manifolds (initially in the differentiable category), including results such as the $$h$$-cobordism theory of Smale (1960), the classification of exotic spheres by Kervaire and Milnor (1962), Browder's converse to the Hirzebruch signature theorem for the existence of a manifold in a simply connected homotopy type (1962), the $$s$$-cobordism theorem of Barden, Mazur, and Stallings (1964), Novikov's proof of the topological invariance of the rational Pontrjagin classes of differentiable manifolds (1965), the fibering theorems of Browder and Levine (1966) and Farrell (1967), Sullivan's exact sequence for the set of manifold structures within a simply connected homotopy type (1966), Casson and Sullivan's disproof of the Hauptvermutung for piecewise linear manifolds (1967), Wall's classification of homotopy tori (1969), and Kirby and Siebenmann's classification theory of topological manifolds (1970).

The original edition of the book fulfilled five purposes by providing:

. a coherent framework for relating the homotopy theory of manifolds to the algebraic theory of quadratic forms, unifying many of the previous results;

. a surgery obstruction theory for manifolds with arbitrary fundamental group, including the exact sequence for the set of manifold structures within a homotopy type, and many computations;

. the extension of surgery theory from the differentiable and piecewise linear categories to the topological category;

. a survey of most of the activity in surgery up to 1970;

. a setting for the subsequent development and applications of the surgery classification of manifolds.

This new edition of this classic book is supplemented by notes on subsequent developments. References have been updated and numerous commentaries have been added. The volume remains the single most important book on surgery theory.

Graduate students and research mathematicians working in the algebraic and geometric topology of manifolds.

Preliminaries
• Note on conventions
• Basic homotopy notions
• Surgery below the middle dimension
• Appendix: Applications
• Simple Poincaré complexes
The main theorem
• Statement of results
• An important special case
• The even-dimensional case
• The odd-dimensional case
• The bounded odd-dimensional case
• The bounded even-dimensional case
• Completion of the proof
Patterns of application
• Manifold structures on Poincaré complexes
• Applications to submanifolds
• Submanifolds: Other techniques
• Separating submanifolds
• Two-sided submanifolds
• One-sided submanifolds
Calculations and applications
• Calculations: Surgery obstruction groups
• Calculations: The surgery obstructions
• Applications: Free actions on spheres; General remarks
• An extension of the Atiyah-Singer $$G$$-signature theorem
• Free actions of $$S^1$$
• Fake projective spaces (real)
• Fake lens spaces
• Applications: Free uniform actions on euclidean space
• Fake tori
• Polycyclic groups
• Applications to 4-manifolds
Postscript
• Further ideas and suggestions: Recent work; Function space methods
• Topological manifolds
• Poincaré embeddings
• Homotopy and simple homotopy
• Further calculations
• Sullivan's results
• Reformulations of the algebra
• Rational surgery
• References
• Index