Lusternik-Schnirelmann invariants in proper homotopy theory

Abstract

We introduce and study proper homotopy invariants of the Luster- nik-Schnirelmann type, p-cat (-), p-Cat(-), and cat e(-) in the cat- egory of Γ2-locally compact spaces and proper maps. As an applica- tion, Rn (n Φ 3) is characterized as (i) the unique open manifold X with p-Cat(ΛΓ) = 2, or (ii) the unique open manifold with one strong end and p-cat( c) = 2. Introduction. The category cat(JΓ) of a space X in the sense of Lusternik and Schnirelmann (L-S category) is the smallest number k such that there exists an open covering {X\, ... , Xk} of X for which each inclusion Xj c X is nullhomotopic in X. This concept was introduced by the quoted authors in their studies on calculus of variations (16) and they used it as a lower bound for the number of critical points of a differentiable real function on a manifold. The basic work on the homotopical significance of cat is due to Borsuk (see (5)). Borsuk's work was continued by Fox (10). Here we present the definition and the basic properties of a new numerical topological invariant for Γ2-locally compact spaces which agrees with the notion of L-S category for ^-compact spaces. This in- variant, denoted p-cat(X), is called the proper L-S category of X and turns out to be a proper homotopy invariant of X. Hence, p-cat(X) is a finer invariant than cat(X). In (10) several generalizations of L-S category are suggested. More explicitly, a general notion of L-S si -category with respect to a class si of spaces has been developed by Puppe and Clapp in (6). Our work shares some common points with (6) but does not fit into the notion of L-S si -category since we entirely deal with proper maps instead of ordinary continuous maps. Another generalization of L-S category has been given in (1), where L-S category for pro-objects in pro-c%/* is defined. This idea is re- lated to proper L-S category by the Edwards-Hastings embedding (see (8)) which provides a close link between proper homotopy theory and homotopy in