Surgery Theory

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.

“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)

• 1 Introduction.- 2 The s-Cobordism Theorem.- 3 Whitehead Torsion.- 4 The Surgery Step and ξ-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.