| Abstract: | In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted byQcat(X). It is obtained by applying a base-point free version ofQ=Ω∞∑∞ fibrewise to the Ganea fibrations. We provecat(X)≥Qcat(X)≥σcat(X) whereσcat(X) denotes Y. Rudyak’s strict category weight. However,Qcat(X) approximatescat(X) better, because, e.g., in the case of a rational spaceQcat(X)=cat(X) andσcat(X) equals the Toomer invariant. We show thatQcat(X×Y)≤Qcat(X)+Qcat(Y). The invariantQcat is designed to measure the failure of the formulacat(X×S r )=cat(X)+1. In fact for 2-cell complexesQcat(X)<cat(X)⇔cat(X×S r )=cat(X) for somer≥1. We note that the paper is written in the more general context of a functor λ from the category of spaces to itself satisfying certain conditions; λ=Q, Ω n Σ n ,Sp ∞ orL f are just particular cases. |
