are concerned with extensions to a categorical setting of the notions of LS-category,
Hopf invariants, and category weight, respectively.
In addition to these primarily homotopy-theoretic articles, three more give
applications to other fields from a homotopy point of view. The articles of COLMAN,
0PREA, and OPREA-RUDYAK give applications to finite group actions, torus actions
and 3-manifolds, respectively.
We have mentioned that Hopf invariants have played a major role in recent
homotopy-theoretic developments. Somewhat surprisingly perhaps, these have been
connected with dynamical systems as well. Indeed, there is a whole branch con-
sisting of a homotopical approach to dynamics, essentially developing from Morse
theory and the work of Conley. This work linking the homotopical development of
LS-category to dynamics has also spilled over into the world of symplectic topology,
where category and related invariants have proved to be useful tools in investigat-
ing subjects such as Hamiltonian circle actions and the Arnold conjecture (for fixed
Lagrangian intersections). The dynamical viewpoint is represented in
this volume by the article of FARBER. Several of the references in his article give
a starting point from which to delve into this branch. Also in the area of dy-
namical systems, but in a decidedly more classical vein, the article by GAVRILA
gives a version of the original Lusternik-Schnirelmann theorem for a closely-related
Finally, the article by COLMAN-HURDER represents an area in which ideas con-
nected with LS-category have only recently been recognized as important, namely
foliations. The references in that article provide many sources for this promising
area of development. The article, and indeed the appearance of LS-category in
foliations, illustrates very well the kind of cross-fertilization that we hoped to foster
at the conference.
One word about notation: The original definition of LS-category, as given in
the article of HILTON, would yield the LS-category of a sphere as 2. In homotopy
theory, however, it is usual to adjust the definition by 1 in such a way that the
sphere has LS-category equal to
The articles by HILTON, COLMAN, COLMAN-
HURDER, FARBER, and GAVRILA adopt the former convention, while all the other
articles in this volume adopt the latter.
We would like to thank the AMS on several counts. First, we must thank it in a
global way for its financial support of the Summer Research Conferences. In a local
way, we thank it for its financial and administrative support for our conference.
In particular, our on-site administrative staff person was Donna Salter, and it can
safely be ventured that the success of the conference was due in large measure to
her organizational skills. Subsequently, the AMS publications department has been
very encouraging and supportive throughout the preparation of this volume. We
would like especially to thank Christine Thivierge for her guidance at each stage.