"LusternikSchnirelmann category is like a Picasso painting. Looking at category from different perspectives produces completely different impressions of category's beauty and applicability." from the Introduction LusternikSchnirelmann category is a subject with ties to both algebraic topology and dynamical systems. The authors take LScategory as the central theme, and then develop topics in topology and dynamics around it. Included are exercises and many examples. The book presents the material in a rich, expository style. The book provides a unified approach to LScategory, including foundational material on homotopy theoretic aspects, the LusternikSchnirelmann theorem on critical points, and more advanced topics such as Hopf invariants, the construction of functions with few critical points, connections with symplectic geometry, the complexity of algorithms, and category of \(3\)manifolds. This is the first book to synthesize these topics. It takes readers from the very basics of the subject to the state of the art. Prerequisites are few: two semesters of algebraic topology and, perhaps, differential topology. It is suitable for graduate students and researchers interested in algebraic topology and dynamical systems. Readership Graduate students and research mathematicians interested in algebraic topology and dynamical systems. Reviews "Finally! The book that sums up the explosive development of the LjusternikSchnirelman theory in the past decade has now appeared."  Zentralblatt MATH "A carefully written, wellconceived and timely addition to the literature on category ... copious references, many interesting exercises and two helpful appendices ... should prove invaluable both as a reference for experts and as a text for a graduate seminar."  Mathematical Reviews Table of Contents  Introduction to LScategory
 Lower bounds for LScategory
 Upper bounds for category
 Localization and category
 Rational homotopy and category
 Hopf invariants
 Category and critical points
 Category and symplectic topology
 Examples, computations and extensions
 Topology and analysis
 Basic homotopy
 Bibliography
 Index
