On the Lusternik-Schnirelmann Category of Maps

Abstract

AbstractWe give conditions which determine if cat of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a CW complex Z such that cat(Z) = 2 but every skeleton of Z is of category 1. We also find conditions when cat(f × g) < cat(f) + cat(g). We apply our result to show that under suitable conditions for rational maps f, mcat(f) < cat(f) is equivalent to cat(f) = cat(f × idSn). Many examples with mcat(f) < cat(f) satisfying our conditions are constructed. We also answer a question of Iwase by constructing p-local spaces X such that cat(X × S1) = cat(X) = 2. In fact for our spaces and every Y ≄ *, cat(X × Y) ≤ cat(Y) + 1 < cat(Y) + cat(X). We show that this same X has the property cat(X) = cat(X × X) = cl(X × X) = 2.