Surgery on compact manifolds. 2nd ed.

*(English)*Zbl 0935.57003
Mathematical Surveys and Monographs. 69. Providence, RI: American Mathematical Society (AMS). xv, 302 p. (1999).

The first edition of this book (1970; Zbl 0219.57024)] is a standard reference on the methods and applications of surgery on compact manifolds. The manifolds need not be closed, connected, or simply connected. The second edition was prepared by A. A. Ranicki, who writes that the book presents: (1) a coherent framework for relating the homotopy theory of manifolds to the algebraic theory of quadratic forms, unifying many of the previous results; (2) 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; (3) the extension of surgery theory from the differentiable and piecewise linear categories to the topological category; (4) a survey of most of the activity in surgery up to 1970; (5) a setting for the subsequent development and applications of the surgery classification of manifolds.

Wall’s book is not regarded as easy to read. In order to lighten the heavy demands on the reader, Ranicki suggests material that can be omitted in the first reading of the book, and still acquire the foundations of surgery theory. The original text has been unaltered, with the same three parts: Part 1: The main theorem, Part 2: Patterns of application, and Part 3: Calculations and applications. Ranicki has added (in italic type) notes at the beginnings and ends of various chapters, as well as footnotes. He also has updated and renumbered the references, and provided the current terminology where it differs from that used in the first edition.

Readers unfamiliar with surgery theory are referred to the articles of J. W. Milnor [Proc. Sympos. Pure Math. 3, 39-55 (1961; Zbl 0118.18601)] and of M. A. Kervaire and J. W. Milnor [Ann. Math., II. Ser. 77, 504-537 (1963; Zbl 0115.40505)], as well as to the books of two pioneers of the field of surgery, W. Browder [Surgery on simply-connected manifolds, Ergeb. Math. Grenzgeb. 65 (1972; Zbl 0239.57016)] and S. P. Novikov [Topology I, Encycl. Math. Sci. 12, 1-310 (1996; Zbl 0839.55001)]. As Ranicki notes, among the most important developments in surgery since 1970, there are the controlled and bounded theories of F. Quinn [Invent. Math. 72, 267-284 (1983; Zbl 0555.57003); corrigendum ibid. 85, 653 (1986; Zbl 0603.57008)] and of S. C. Ferry and E. K. Pedersen [Epsilon surgery theory, Lond. Math. Soc. Lect. Note Ser. 227, 167-226 (1995)]. Ranicki refers the reader to collections of papers edited by S. E. Cappell, A. Ranicki, and J. Rosenberg [Surveys on surgery theory: Vol. 1, Papers dedicated to C. T. C. Wall, Ann. Math. Stud. 145 (2000)] and S. C. Ferry, A. Ranicki and J. Rosenberg [Novikov conjectures, index theorems and rigidity: Oberwolfach, 1993, Vols. 1, 2, Lond. Math. Soc. Lect. Note Ser. 226, 227 (1995)] for surveys of many areas of surgery theory, subsequent developments, and applications, including the methods used to verify (for a large class of groups) the Novikov and Borel conjectures via the assembly map.

The 1970 first edition of Wall’s book exerted a great influence on the pattern of the methods in surgery theory developed later on. The 1999 second edition, with all of the additions done by Ranicki, should increase the already big interest in this fundamental indispensable book on surgery theory, and attract the attention of topologists for many years to come.

Wall’s book is not regarded as easy to read. In order to lighten the heavy demands on the reader, Ranicki suggests material that can be omitted in the first reading of the book, and still acquire the foundations of surgery theory. The original text has been unaltered, with the same three parts: Part 1: The main theorem, Part 2: Patterns of application, and Part 3: Calculations and applications. Ranicki has added (in italic type) notes at the beginnings and ends of various chapters, as well as footnotes. He also has updated and renumbered the references, and provided the current terminology where it differs from that used in the first edition.

Readers unfamiliar with surgery theory are referred to the articles of J. W. Milnor [Proc. Sympos. Pure Math. 3, 39-55 (1961; Zbl 0118.18601)] and of M. A. Kervaire and J. W. Milnor [Ann. Math., II. Ser. 77, 504-537 (1963; Zbl 0115.40505)], as well as to the books of two pioneers of the field of surgery, W. Browder [Surgery on simply-connected manifolds, Ergeb. Math. Grenzgeb. 65 (1972; Zbl 0239.57016)] and S. P. Novikov [Topology I, Encycl. Math. Sci. 12, 1-310 (1996; Zbl 0839.55001)]. As Ranicki notes, among the most important developments in surgery since 1970, there are the controlled and bounded theories of F. Quinn [Invent. Math. 72, 267-284 (1983; Zbl 0555.57003); corrigendum ibid. 85, 653 (1986; Zbl 0603.57008)] and of S. C. Ferry and E. K. Pedersen [Epsilon surgery theory, Lond. Math. Soc. Lect. Note Ser. 227, 167-226 (1995)]. Ranicki refers the reader to collections of papers edited by S. E. Cappell, A. Ranicki, and J. Rosenberg [Surveys on surgery theory: Vol. 1, Papers dedicated to C. T. C. Wall, Ann. Math. Stud. 145 (2000)] and S. C. Ferry, A. Ranicki and J. Rosenberg [Novikov conjectures, index theorems and rigidity: Oberwolfach, 1993, Vols. 1, 2, Lond. Math. Soc. Lect. Note Ser. 226, 227 (1995)] for surveys of many areas of surgery theory, subsequent developments, and applications, including the methods used to verify (for a large class of groups) the Novikov and Borel conjectures via the assembly map.

The 1970 first edition of Wall’s book exerted a great influence on the pattern of the methods in surgery theory developed later on. The 1999 second edition, with all of the additions done by Ranicki, should increase the already big interest in this fundamental indispensable book on surgery theory, and attract the attention of topologists for many years to come.

Reviewer: K.Pawałowski (Poznań)

##### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

57R65 | Surgery and handlebodies |

57R67 | Surgery obstructions, Wall groups |

19J25 | Surgery obstructions (\(K\)-theoretic aspects) |