Preface The emergence of differential topology and dynamical systems can be traced back to the work of Poincare on analysis situs at the dawn of the 20th century. Poincare's recognition that the existence and form of solutions of differential equa- tions were intimately connected with the "topology" of the space where the equa- tions found their natural definition led to new ideas in analysis and the develop- ment of previously vague notions such as that of "manifold". After these concepts crystalized somewhat, an immediate basic problem was (and still is) to relate the complexity of flows to the topological complexity of the underlying manifold. In this context, a first step was to estimate the number of invariant (or rest) points for the particular case of gradient flows or, equivalently, to estimate the minimal number of critical points of functions on the manifold. Morse's work in the late 20's and early 30's led to such estimates for particular generic functions: those whose critical points were non-degenerate. Around the same time, L. Lusternik and L. Schnirelmann ([LS34]) described a new invariant of a manifold called category. Their aim in creating this notion was to provide a lower bound on the number of critical points for any smooth function on the manifold. While this aim was analytical in nature, it had far-reaching consequences in geometry as well. As we shall see later (see Theorem 9.12), the general approach of Lusternik and Schnirelmann can be applied to obtain results such as the existence of a closed geodesic. Indeed, Lusternik and Schnirelmann were able to use their new invariant to prove wonderful results such as the existence of three closed geodesies on the sphere. Furthermore, once reformulated by Fox ([Fox41a]), category (or Lusternik-Schnirelmann category as it became known) found a useful niche in algebraic topology. For example, the category of a space X was used by G. Whitehead to bound from above the nilpotency class of the group of homotopy classes from X to a group-like space ([Whi54]). Thus began a long association of category with the notion of nilpotency. Category continued to be a tool in critical point theory, but it also became a main focus of the "numerical invariant" movement in homotopy theory in the 50's. After foundational results were obtained in the 1960's, the problem list of T. Ganea ([Gan71]) served to motivate further study of category by topologists. The development of localization techniques in topology and, particularly, the creation of Sullivan's version of rational homotopy theory spawned new approximating invari- ants which energized the field and which led to greater understanding in areas as diverse as the study of the homotopy Lie algebra ([Fel89]) and the number of fixed points for certain diffeomorphisms on some types of manifolds (see Theorem 8.28). Recently, new approximating invariants for category have been successfully em- ployed to solve an example of the latter problem called the Arnold conjecture for symplectic manifolds ([Rud99a], [R099]). Simultaneously, some of the recent,
Previous Page Next Page