Sectional Category of the Ganea Fibrations and Higher Relative Category.

We first compute James’ sectional category (secat) of the Ganea map gk of any map ιX in terms of the sectional category of ιX: we show that secat gk is the integer part of secat ιX/(k+1). Next we compute the relative category (relcat) of gk. In order to do this, we introduce the relative category of order k (relcatk) of a map and show that relcat gk is the integer part of relcatkιX/(k+1). Then we establish some inequalities linking secat and relcat of any order: we show that secat ιX⩽relcatkιX⩽secat ιX+k+1 and relcatkιX⩽relcatk+1ιX⩽relcatkιX+1. We give examples that show that these inequalities may be strict. [ABSTRACT FROM AUTHOR]