Lie structures in homotopy and isotopy calculi

TL;DR

This paper explores Lie algebra structures in homotopy and isotopy calculus, providing new proofs, comparisons, and geometric definitions to unify these frameworks through a novel bracket construction.

Contribution

It introduces a new proof of the Johnson--Arone--Mahowald result and defines a geometric bracket on embedding calculus layers, unifying these approaches via a novel bracket on total homotopy fibres.

Findings

Spectral Lie bracket compared with Whitehead bracket

New proof of Goodwillie tower layers

Geometric bracket on embedding calculus layers

Abstract

We establish compatibility of Lie structures that appear in homotopy calculus of functors and isotopy calculus of embeddings. On one hand, we give a new proof of the Johnson--Arone--Mahowald result describing the layers of the Goodwillie tower of the identity functor, and we directly compare the spectral Lie bracket with the classical Whitehead bracket on spaces. On the other hand, we geometrically define a bracket on the layers of the embedding calculus tower for embeddings of arcs. These results are unified through the same technical tool, a newly defined bracket on total homotopy fibres of collapsing cubes of wedge sums.