Yet another Hopf Invariant

TL;DR

The paper defines the Hopf category $hcat(f)$ for maps $f: \Sigma_A W \to X$ via the Ganea construction, linking Hopf-type invariants with Lusternik–Schnirelmann type invariants such as secat and relcat. It establishes fundamental bounds ${\rm secat}(f)\leq {\rm relcat}(\iota_X)\leq {\rm hcat}(f)\leq {\rm relcat}_1(\iota_X)\leq {\rm relcat}(\iota_X)+1$ and shows how $hcat(f)$ behaves under homotopy pushouts, including a lower bound in terms of cofibre category. The authors introduce a strong version ${\rm Hcat}(f)$ and prove ${\rm Relcat}(f)\le {\rm Hcat}(f)$, yielding ${\rm hcat}(f)\ge {\rm relcat}(f)$ and a criterion: if ${\rm relcat}(f)>{\rm relcat}(\iota_X)$ then ${\rm hcat}(f)={\rm relcat}(\iota_X)+1$. They provide equivalent conditions for bounding $hcat$ with prism/pullback arguments and give explicit examples, including Hopf maps, illustrating the attained values (e.g., ${\rm hcat}=2$), thereby unifying Hopf invariants with Ganea calculus for relative suspensions.

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).

Theorems & Definitions (32)

• Definition 1
• Definition 2
• Definition 3
• Example 4
• Example 5
• Proposition 6
• proof
• Example 7
• Proposition 8
• proof