Higher-dimensional knots according to Michel Kervaire (Q2364835)

This book presents Kervaire's work on higher-dimensional knots, with the aim of making the reading of papers by Michel Kervaire and Jerome Levine easier. Here, a knot \(K^n \subset S^{n+q}\) in codimension \(q\) is the image of a differentiable embedding of an \(n\)-dimensional homotopy sphere in the \((n+q)\)-sphere \(S^{n+q}\). In particular, we consider the case \(q=2\). \newline Chapter 2 presents some background in differentiable topology including vector bundles, the Pontrjagin method/construction and the \(J\)-homomorphism, and surgery. Chapter 3 is a summary of Kervaire-Milnor's paper on the study of homotopy spheres, including the Kervaire invariant and the Kervaire manifold. Chapter 4 discusses knots in codimension equal to or greater than 3. Chapter 5 presents Kervaire's determination of the fundamental groups of knot complements in higher dimensions. Chapter 6 presents Kervaire's results and several developments due to Levine on knot modules, Seifert hypersurfaces, Seifert forms and simple knots. Chapter 7 presents Levine's work on odd-dimensional simple knots. Chapter 8 treats higher-dimensional knot cobordism. Chapter 9 summarizes some results on the topological type of singularities of complex hypersurfaces, related to Kervaire's work, on Seifert forms, knot cobordism, etc. The book includes several remarks on the history. The appendices A-E can be read independently. The appendices A-D present basics about linking numbers and signs, the existence of Seifert hypersurfaces, open book decompositions, handlebodies and plumbings. Appendix E presents the result due to Hill-Hopkins-Ravenel on the Kervaire invariant problem and its consequences for higher-dimensional knot theory.