About sectional category of the Ganea maps
Abstract
We first compute the James' sectional category (secat) of the Ganea map gk of any map f in terms of the sectional category of f: We show that secat(gk) is the integer part of secat(f)/(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 relcatk(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcatk(f) <= secat(f) + k + 1 and relcatk(f) <= relcat(k+1)(f) <= relcatk(f) + 1. We give examples that show that these inequalities may be strict.
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