About sectional category of the Ganea maps

2015/07/28 by Doeraene, Jean-Paul
#55M30 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.07752

Abstract

We first compute the James' sectional category (secat) of the Ganea map g_k of any map f in terms of the sectional category of f: We show that secat(g_k) is the integer part of secat(f)/(k+1). Next we compute the relative category (relcat) of g_k. In order to do this, we introduce the relative category of order k (relcat_k) of a map and show that relcat(g_k) is the integer part of relcat_k(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcat_k(f) <= secat(f) + k + 1 and relcat_k(f) <= relcat_(k+1)(f) <= relcat_k(f) + 1. We give examples that show that these inequalities may be strict.

Related

• The Ganea conjecture for rational approximations of sectional category
• Up to one approximations of sectional category and topological complexity
• The sectional category of a map
• Relative sectional category revisited
• The rational sectional category of certain maps