Title: An invitation to Morse theory /
Authors: Nicolaescu, Liviu I.
Publisher: Springer,
Year: 2007
Subjects: Morse theory.; Mathematics.; Global analysis (Mathematics); Cell aggregation Mathematics.
Abstract: "This self-contained treatment of Morse Theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory (over the reals). The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.k.a. Picard-Lefschetz theory. This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations, and will be of interest to graduate students as well as researchers. The reader is expected to have some familiarity with cohomology theory and with the differential and integral calculus on smooth manifolds"--Publisher description.
Contents: Notations and conventions -- 1. Morse functions -- The local structure of Morse functions -- Existence of Morse functions -- 2. The topology of Morse functions -- Surgery, handle attachment, and cobordisms -- The topology of sublevel sets -- Morse inequalities -- Morse-Smale dynamics -- Morse-Floer homology -- Morse-Bott functions -- Min-Max theory -- 3. Applications -- The cohomology of complex grassmannians -- Lefschetz hyderplane theorem -- Symplectic manifolds and Hamiltonian flows -- Morse theory of moment maps -- S¹-Equivariant localization -- 4. Basics of complex Morse theory -- Some fundamental constructions -- Topological applications of Lefschetz pencils -- The hard Lefschetz theorem -- Vanishing cycles and local monodromy -- Proof of the Picard-Lefschetz formulæ -- 5. Exercises and solutions -- Exercises -- Solutions to selected exercises.