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This monograph provides a comprehensive introduction to surgery theory, the main tool in the classification of manifolds.
Surgery theory was developed to carry out the so-called Surgery Program, a basic strategy to decide whether two closed manifolds are homeomorphic or diffeomorphic. This book provides a detailed explanation of all the ingredients necessary for carrying out the surgery program, as well as an in-depth discussion of the obstructions that arise. The components include the surgery step, the surgery obstruction groups, surgery obstructions, and the surgery exact sequence. This machinery is applied to homotopy spheres, the classification of certain fake spaces, and topological rigidity. The book also offers a detailed description of Ranicki's chain complex version, complete with a proof of its equivalence to the classical approach developed by Browder, Novikov, Sullivan, and Wall.
This book has been written for learning surgery theory and includes numerous exercises. With full proofs and detailed explanations, it also provides an invaluable reference for working mathematicians. Each chapter has been designed to be largely self-contained and includes a guide to help readers navigate the material, making the book highly suitable for lecture courses, seminars, and reading courses.
List of contents
1 Introduction.- 2 The s-Cobordism Theorem.- 3 Whitehead Torsion.- 4 The Surgery Step and -Bordism.- 5 Poincaré Duality.- 6 The Spivak Normal Structure.- 7 Normal Maps and the Surgery Problem.- 8 The Even-Dimensional Surgery Obstruction.- 9 The Odd-Dimensional Surgery Obstruction.- 10 Decorations and the Simple Surgery Obstruction.- 11 The Geometric Surgery Exact Sequence.- 12 Homotopy Spheres.- 13 The Geometric Surgery Obstruction Group and Surgery Obstruction.- 14 Chain Complexes.- 15 Algebraic Surgery.- 16 Brief Survey of Computations of L-Groups.- 17 The Homotopy Type of G/TOP, G/PL, and G/O.- 18 Computations of Topological Structure Sets of some Prominent Closed Manifolds.- 19 Topological Rigidity.- 20 Modified Surgery.- 21 Solutions of the Exercises.
About the author
Wolfgang Lück is a C3-Professor of mathematics at the University of Bonn. Prior to Bonn, he worked as professor at the universities of Lexington, Mainz, and Münster. He obtained his PhD in 1984 at the University of Göttingen. His main research areas are topology,
K- and
L-theory, and
L2-invariants, and has more than 140 research publications. He has supervised about 30 PhD-students. He is a recipient of the Max Planck Research Award, the Gottfried Wilhelm Leibniz Prize, and an ERC Advanced Grant. He is a member of the German National Academy of Sciences Leopoldina and of the North Rhine-Westphalian Academy of Sciences, Humanities and Arts, and is a Fellow of the American Mathematical Society.
Tibor Macko is an associate professor of mathematics at the Comenius University and a research fellow at the Slovak Academy of Sciences, both in Bratislava. He obtained his PhD in 2004 from the University of Aberdeen and was later apostdoc at MPI in Bonn and at the universities at Münster and Bonn before returning to his native Slovakia. He is an author or a coauthor of more than a dozen papers on topology of manifolds.
Report
It is nice that so much about surgery is collected in one volume. The book brings together topics that are scattered over the literature, and sometimes difficult to find. Even the experienced reader may find something new. The extensive bibliography is great to have. (Karl Heinz Dovermann, zbMATH 1551.57001, 2025)
