\(A_{\infty }\)-spaces and L-S category in the category of fibrewise spaces (Q989094)

Fibrewise homotopy theory has been introduced by James and was developed by \textit{M. Crabb} and \textit{I. James} in their book [Fibrewise homotopy theory. London: Springer (1998; Zbl 0905.55001)]. Then, in [Proc. R. Soc. Edinb., Sect. A 119, No.~1--2, 177--190 (1991; Zbl 0738.55005)], \textit{I. M. James} and \textit{J. R. Morris} defined the LS category for fibrewise pointed spaces. In the present paper the author continues the study of the fibrewise LS category. He constructs a fibrewise version of the projective spaces for A\(_\infty\)-spaces, i.e., fibrewise versions of the Ganea fibrations associated to a space, and he proves the fibrewise analogue of the classical theorem of Ganea: The LS category of a space \(X\) is \(\leq n\) if and only if the \((n+1)^{th}\) Ganea fibration associated to \(X\) has a section.