**Mathematical Surveys and Monographs**

Volume: 69;
1999;
302 pp;
Hardcover

MSC: Primary 57;
Secondary 18; 19; 11

Print ISBN: 978-0-8218-0942-6

Product Code: SURV/69

List Price: $72.00

Individual Member Price: $57.60

**Electronic ISBN: 978-1-4704-1296-8
Product Code: SURV/69.E**

List Price: $72.00

Individual Member Price: $57.60

#### Supplemental Materials

# Surgery on Compact Manifolds: Second Edition

Share this page *Edited by *
*A. A. Ranicki*

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.

#### Table of Contents

# Table of Contents

## Surgery on Compact Manifolds: Second Edition

- Contents vii8 free
- Forewords ix10 free
- Editor's foreword to the second edition xi12 free
- Introduction xv16 free
- Part 0: Preliminaries 118 free
- Part 1: The main theorem 3148
- Part 2: Patterns of application 105122
- Part 3: Calculations and applications 171188
- 13A. Calculations: surgery obstruction groups 172189
- 13B. Calculations: the surgery obstructions 185202
- 14. Applications: free actions on spheres 195212
- 14A. General remarks 195212
- 14B. An extension of the Atiyah-Singer G-signature theorem 199216
- 14C. Free actions of S[sup(1)] 202219
- 14D. Fake projective spaces (real) 206223
- 14E. Fake lens spaces 213230
- 15. Applications: free uniform actions on euclidean space 231248
- 15A. Fake tori 232249
- 15B. Polycyclic groups 237254
- 16. Applications to 4-manifolds 241258

- Part 4: Postscript 249266
- References 285302
- Index 300317 free

#### Readership

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