Yet another Hopf invariant

Abstract

The classical Hopf invariant is defined for a map \(f\colon S^r \to X\). Here we define 'hcat' which is some kind of Hopf invariant built with a construction in Ganea's style, valid for maps not only on spheres but more generally on a 'relative suspension' \(f\colon \Sigma_A W \to X\). We study the relation between this invariant and the sectional category and the relative category of a map. In particular, for \(\iota_X\colon A\to X\) being the 'restriction' of \(f\) on \(A\), we have \({\rm relcat}~\iota_X \leq {\rm hcat}~f \leq {\rm relcat}~ \iota_X +1\) and \({\rm relcat}~f \leq {\rm hcat}~f\).