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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.
Table des matières
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.
A propos de l'auteur
Wolfgang Lück has worked on topology, K-theory, and global analysis. He completed his PhD in 1984 under the supervision of Prof. Tammo tom Dieck at Göttingen, where he also obtained the venia legendi in 1989. He has held permanent positions at the universities at Lexington, Mainz and Münster, and is currently Professor at the University of Bonn. He was awarded the Max Planck Research Award in 2003, the Gottfried Wilhelm Leibniz Award in 2008 and the von Staudt Prize in 2025. His other honors include membership of the Leopoldina (since 2010) and of the Nordrhein-Westfälische Akademie der Wissenschaft und der Künste (since 2013), Fellowship of the American Mathematical Society (since 2013), and he was president of the Deutsche Mathematiker Vereinigung during 2009–2010. In addition, he was a Max Planck Fellow from 2013–2023 and obtained an ERC Advanced Grant in 2014. To date, he has directed the theses of 30 PhD students. In Bonn, he was the director of the Hausdorff Institute from 2011–2017 and the spokesperson of the Cluster of Excellence Hausdorff Center for Mathematics from 2019–2022. He has been married to Sibylle Lück since 1984 and has four children and four grandchildren.
Commentaire
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)
