Finally! The book that sums up the explosive development of the Ljusternik-Schnirelman theory in the past decade has now appeared. It is also worth to mention the following conference proceedings: [Cornea, O. (ed.); Lupton, G.(ed.); Oprea, J.(ed.); Tanré, D.(ed.), Lusternik-Schnirelmann category and related topics. Proceedings of the 2001 AMS-IMS-SIAM joint summer research conference on Lusternik-Schnirelmann category in the new millennium, South Hadley, MA, USA, July 29–August 2, 2001 (Zbl 1005.00038)].
The Ljusternik-Schnirelman (in future LS-) category \(\text{ cat } X\) of a topological space \(X\) is defined to be the minimal number \(k\) such that \(X\) can be covered by \(k+1\) open and contractible (in \(X\)) sets. Ljusternik and Schnirelman introduced this invariant for manifolds and proved that every function on a closed smooth manifold \(M\) has at least cat \(M\)+1 critical points. Later Borsuk and Fox suggested to consider cat \(X\) as an invariant of topological spaces. It turns out that cat \(X\) is a homotopy invariant. Nevertheless, it is quite complicated to compute cat \(X\). Being at the periphery of topology in the mid of last century, the LS theory is now one of central directions of algebraic and geometric topology. Many problems of the theory were solved quite recently, and several new interesting applications were discovered.
The reviewed book gathers most of these results under the same cover. Sometimes the exposition is quite close to original papers, this happens because the authors include up-to-date results that are not published yet and exist in manuscript or electronic form only. The book contains exercises that are chosen with a good taste, and some open problems.
The contents of the book are the following:
Ch. 1. Introduction to LS-Category. Here the authors give the definition and state basic properties of LS-category and give homotopy interpretations of LS category via the Whitehead fat wedge and the Ganea-Svarz fibration.
Ch. 2. Lower bounds for LS-Category. Here the authors consider some invariants that estimate LS-category from below (cup-length, conilpotency, Toomer invariant, \(\sigma\) cat, weak category) and give some comparison results for these invariants. The relatively new concept of category weight is also treated in the chapter.
Ch. 3. Upper bounds for Category. In this chapter the authors compare the category with other numerical covering invariants (strong category, cone-length, covering of manifolds by balls) and consider some cases when the category is equal to these invariants.
Ch. 4. Localization and Category. Here the authors explain backgrounds on localization and prove some relations between the category of a space and the localized space. Also, in this chapter, behaviour of the category under fiberwise constructions is considered.
Ch. 5. Rational Homotopy and Category. Here the authors describe the category in terms of Sullivan minimal models and consider related rational invariants (cat\(_0\), mcat).
Ch. 6. Hopf Invariants. The concepts of this chapter is concentrated about the following question: what happens with the category of a space after attaching a cell? Most results say that the category does not increase provided that a suitably defined Hopf invariant of an attaching map is equal to zero. Also, in this chapter the authors explain Iwase’s counterexample to the Ganea conjecture.
Ch. 7. Category and Critical Points. Here the authors explain the Conley index theory, discuss some results on upper bounds for minimal numbers of critical points of functions on manifolds, develop analogs of LS-theory for manifolds with boundary and open manifolds, provided that the functions have controlled behavior on the boundary or at infinity.
Ch. 8. Category and Symplectic Topology. Here the authors explain applications of LS-theory to the Arnold conjecture (fixed points of Hamiltonian symplectomorphisms, Lagrangian intersections) and to symplectic group actions.
Ch. 9. Examples, Computations and Extensions. Several interesting topics, including Smale’s application of LS-theory to the complexity of algorithms.