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 highdimensional 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. Readership Graduate students and research mathematicians working in the algebraic and geometric topology of manifolds. Table of Contents 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 evendimensional case
 The odddimensional case
 The bounded odddimensional case
 The bounded evendimensional case
 Completion of the proof
Patterns of application  Manifold structures on Poincaré complexes
 Applications to submanifolds
 Submanifolds: Other techniques
 Separating submanifolds
 Twosided submanifolds
 Onesided submanifolds
Calculations and applications  Calculations: Surgery obstruction groups
 Calculations: The surgery obstructions
 Applications: Free actions on spheres; General remarks
 An extension of the AtiyahSinger \(G\)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 4manifolds
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
