On the James and Hilton-Milnor Splittings, & the metastable EHP sequence

TL;DR

This paper provides modern, general proofs of classical algebraic topology results like the James and Hilton-Milnor splittings and the metastable EHP sequence, extending their validity to broader contexts such as $$-topoi and motivic spaces.

Contribution

It introduces a unified, abstract framework for these splittings using $$-categories, enabling their application in new mathematical settings and offering novel proofs of the EHP sequence.

Findings

Splittings hold in elementary $$-topoi, profinite, and motivic spaces.

New non-computational proof of the metastable EHP sequence.

Extensions of classical results to broader categorical contexts.

Abstract

This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results \begin{equation*} \Sigma \Omega \Sigma X \simeq \Sigma X \vee (X\wedge \Sigma\Omega \Sigma X) \quad \text{and} \quad \Omega(X \vee Y) \simeq \Omega X\times \Omega Y\times \Omega \Sigma(\Omega X \wedge \Omega Y) \end{equation*} in the maximal generality of an $\infty$-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor Splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary $\infty$-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting result in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of $\infty$-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof.