Preface This collection of articles is the proceedings volume for the American Mathe- matical Society's Summer Research Conference, Lusternik-Schnirelmann Category in the New Millennium, held 29th July-2nd August 2001 on the campus of Mount Holyoke College in Massachusetts. The conference, one of seven joint AMS-IMS- SIAM summer research conferences in the mathematical sciences held that summer, attracted an international group of 37 participants that included many of the lead- ing practitioners in the field. Lusternik-Schnirelmann category (LS-category) is an integer that can be asso- ciated to a topological space. It is an invariant of the homotopy type of the space that gives a numerical measure of the complexity of the space. In particular, it indicates the complexity of possible dynamics on a smooth· manifold, by providing a lower bound on the number of critical points of any smooth function on the man- ifold. The survey article by HILTON in this volume recalls the basic definitions and those results that first brought category to the attention of topologists. While LS-category is classical in origin, the subject has recently enjoyed a renaissance. The latest developments include work in the areas of homotopy theory, dynamical systems, and symplectic topology. One interesting aspect of this recent activity is the way in which it has made significant links connecting these areas. Many of the new developments have occurred in the homotopy-theoretic branch of the subject. The composition of the articles in this volume reflects this fact. Of the fifteen contributed articles, nine are primarily homotopy-theoretic. The survey article by HILTON gives a resume of those homotopy-theoretic results known before the surge of recent activity. While several of the contributed articles continue in this classical vein, practically all of them are influenced in one way or another by more recent ideas. Certainly all of them represent non-classical points of view on LS-category. Broadly speaking, the main forces responsible for spurring new interest in LS- category among homotopy theorists have been developments concerning the Ganea conjecture (that cat(X x sn) = cat(X) + 1) on the one hand and rational homotopy theory on the other. Concerning the Ganea conjecture, there are two ideas that have played a particularly central role in this work. One is the notion of category weight and the other is the notion of Hopf invariant, in both a classical and an extended sense. The articles of STROM and 0PREA-RUDYAK are concerned with category weight, those of DULA and MARCUM with Hopf invariants. The arti- cles by CuviLLIEZ-FELIX and LUPTON are concerned with rational homotopy the- ory proper, and those of COSTOYA-RAMOS, GHIENNE, and HUBBUCK-IWASE with more general localization and completion. Finally, among the primarily homotopy- theoretic articles, those of ARKOWITZ-STANLEY-STROM, MARCUM, and STROM vii
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