Yet another Hopf Invariant

TL;DR

This paper introduces a generalized Hopf invariant called 'hcat' for maps beyond spheres, relating it to sectional and relative categories, and establishing bounds and inequalities among these invariants.

Contribution

It defines the 'hcat' invariant for maps on relative suspensions, extending classical Hopf invariants and connecting it with sectional and relative categories.

Findings

hcat is bounded by relcat(i) and relcat(i)+1

relcat(f) is less than or equal to hcat(f)

hcat provides new insights into the topology of maps beyond spheres

Abstract

The classical Hopf invariant is defined for a map f: S^r -> 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: Sigma_A W -> X. We study the relation between this invariant and the sectional category and the relative category of a map. In particular, for f being the `restriction' of f on A, we have relcat(i) <= hcat(f) <= relcat(i) + 1 and relcat(f) <= hcat(f).