On the Lusternik-Schnirelmann Category

Abstract

In their study of the calculus of variations in the large, Lusternik and Schnirelmann were led to introduce the notion of of a topological space X. Borsuk then began to investigate the relationship between the category of X and the more popular homotopy-type invariants of X. More recent results in this direction are due mainly to R. H. Fox, G. W. Whitehead, P. Hilton, and T. Ganea, but much remains unknown. In this paper we will exhibit some relationships between the category of X and certain spectral sequences which arise from the bar construction of Eilenberg-MacLane [3] and Milnor's universal bundle construction [7]. In particular, if the category of X is k, then the differentials dr as well as many of the groups Er of these sequences are trivial for r>k. As applications of these results, we obtain an exact sequence relating the homology suspension [11] to the Pontrjagin product in I and in ? 3, we demonstrate its equivalence with the bar construction and Milnor's construction. In ? 4, we give a theorem about Massey homology products in &2X and, as an example, compute some of these products in &7CPn, where CPn is complex projective n space. The main results are listed, as theorems and corollaries, at the beginnings of ?? 2, 3, and 4.