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Lie structures in homotopy and isotopy calculi

Danica Kosanović

TL;DR

The paper develops a unified framework for Lie structures in both homotopy and isotopy calculi, showing that the layers of the Goodwillie tower for the identity and the embedding calculus tower for long arcs carry compatible Lie brackets. It introduces a bracket on total homotopy fibres of collapsing cubes that links external Samelson, grafting, and operadic Lie brackets, with a Johnson-type map J providing a bridge between the calculi. The main technical achievement is a precise compatibility diagram between these brackets and maps, yielding a high-connectivity correspondence between Samelson-type operations and Lie operations across settings. This bridge yields new insights into the role of nilpotence in the calculus towers and produces explicit, embedded realizations (grasper classes) of the first nontrivial homotopy groups of embedding layers, with broad implications for both homotopy theory and isotopy theory of embeddings.

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.

Lie structures in homotopy and isotopy calculi

TL;DR

The paper develops a unified framework for Lie structures in both homotopy and isotopy calculi, showing that the layers of the Goodwillie tower for the identity and the embedding calculus tower for long arcs carry compatible Lie brackets. It introduces a bracket on total homotopy fibres of collapsing cubes that links external Samelson, grafting, and operadic Lie brackets, with a Johnson-type map J providing a bridge between the calculi. The main technical achievement is a precise compatibility diagram between these brackets and maps, yielding a high-connectivity correspondence between Samelson-type operations and Lie operations across settings. This bridge yields new insights into the role of nilpotence in the calculus towers and produces explicit, embedded realizations (grasper classes) of the first nontrivial homotopy groups of embedding layers, with broad implications for both homotopy theory and isotopy theory of embeddings.

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.
Paper Structure (33 sections, 20 theorems, 87 equations, 1 figure)

This paper contains 33 sections, 20 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Homotopy calculus
  3. Isotopy calculus
  4. A bridge
  5. Background
  6. Trees
  7. Lie trees
  8. Partition complexes
  9. Grafting
  10. Goodwillie partition complexes
  11. The cooperad structure
  12. Grafting brackets
  13. Samelson brackets and canonical maps
  14. Commutator brackets
  15. Samelson brackets
  16. ...and 18 more sections

Key Result

Theorem 1

For any based space $X$, the structure of a spectral Lie algebra on the layers of the homotopy Taylor tower of the identity functor at $X$ induces brackets that correspond to Samelson brackets: there are maps $w_{D_n}$ that extend $w(x_i)$ from eq-intro:can-Sam, so that if $w=[w_1,w_2]$ with $w_b\in\mathop{\mathrm{\mathsf{Lie}}}\nolimits(S_b)$ for $S_1\sqcup S_2=\ul{n}$, then the following square

Figures (1)

  • Figure 1: The manifold $M_T$, for $T=\{0\}$ and $T=\{0,n+1\}$ respectively, is the complement of the blue material. An element of $\mathop{\mathrm{Emb}}\nolimits_\partial(J_0,M_{0S})$ is an arc embedded in $M_{0S}$ with endpoints like $J_0$. The map $\rho_S^k$ for $S=\emptyset$ and $k=n+1$ is simply the inclusion of the arcs as in the top picture to the space of arcs in the bottom picture.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2: ConantRobinson
  • Theorem 2.3: BarceloAK-LieWachsRobinson
  • proof : Proof Idea
  • Definition 2.4
  • Definition 2.5
  • ...and 39 more