Invariants of the Lusternik - Schnirelmann Type and the Topology of Critical Sets

Abstract

We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function. We also improve a result of T. Ganea about the equality of the strong category and the category (even in the classical case). The category cat( A) of a space A in the sense of Lusternik and Schnirelmann [14] is the smallest number k such that there exists a covering {A,,..., Xk} of A" (of a certain kind, cf. 1.2(1)) for which each inclusion A c A is nullhomotopic. The motivation for introducing this concept was that it gives a lower bound for the number of critical points of a function. More precisely, if M is a closed differentiable manifold and / is a differentiable real function on M then the number of critical points of / is at least cat(M). We propose the following generalization: If si is any class of spaces we replace the condition that Xj c A is nullhomotopic by requiring that it factors through some A g si up to homotopy and we obtain the notion of .«¿category, si-oat(X). If si consists only of the one-point space, this is the classical cat( A). If sé is the class of a-connected spaces si-oat is the "a-dimensional homotopy category" introduced by Fox in [7]. Another interesting example is the class si of a-dimensional spaces. If /: M -> R is as above then si-cat(M) does not give any new information about the number of critical points, because it is less than or equal to cat( M). It does give, however, under certain conditions, some new information on the topological structure of the critical set. Roughly, one can say that either there are at least si-cat(M) critical values of / or there is one critical value y of / such that the corresponding set of critical points K n f~x(y) is not of the homotopy type of any space in si (cf. §2 for more details). Quite a number of papers have appeared on Lusternik-Schnirelmann category and related notions (cf. [13] for a survey). In particular there has been a revival of interest in recent years. We shall present a systematic theory for our more general Received by the editors November 25, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 55M30; Secondary 58E05, 55P50.

Cite this paper

@inproceedings{Puppe2010InvariantsOT, title={Invariants of the Lusternik - Schnirelmann Type and the Topology of Critical Sets}, author={Dieter Puppe}, year={2010} }