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Surgery on Compact Manifolds: Second Edition

C. T. C. Wall University of Liverpool, Liverpool, England
Edited by: A. A. Ranicki University of Edinburgh, Edinburgh, Scotland
Available Formats:
Hardcover ISBN: 978-0-8218-0942-6
Product Code: SURV/69
List Price: $77.00 MAA Member Price:$69.30
AMS Member Price: $61.60 Electronic ISBN: 978-1-4704-1296-8 Product Code: SURV/69.E List Price:$72.00
MAA Member Price: $64.80 AMS Member Price:$57.60
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $115.50 MAA Member Price:$103.95
AMS Member Price: $92.40 Click above image for expanded view Surgery on Compact Manifolds: Second Edition C. T. C. Wall University of Liverpool, Liverpool, England Edited by: A. A. Ranicki University of Edinburgh, Edinburgh, Scotland Available Formats:  Hardcover ISBN: 978-0-8218-0942-6 Product Code: SURV/69  List Price:$77.00 MAA Member Price: $69.30 AMS Member Price:$61.60
 Electronic ISBN: 978-1-4704-1296-8 Product Code: SURV/69.E
 List Price: $72.00 MAA Member Price:$64.80 AMS Member Price: $57.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$115.50 MAA Member Price: $103.95 AMS Member Price:$92.40
• Book Details

Mathematical Surveys and Monographs
Volume: 691999; 302 pp
MSC: Primary 57; Secondary 18; 19; 11;

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.

• Chapters
• Part 0. Preliminaries
• Part 1. The main theorem
• Part 2. Patterns of application
• Part 3. Calculations and applications
• Part 4. Postscript

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Accessibility – to request an alternate format of an AMS title
Volume: 691999; 302 pp
MSC: Primary 57; Secondary 18; 19; 11;

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.