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The algebra of higher homotopy operations

Samik Basu, David Blanc, Debasis Sen

TL;DR

This work develops a coherent algebraic framework for simplicial higher order unstable homotopy operations, formulated in a model-free ∞-categorical setting, and connects these operations to spectral sequence indeterminacies. It clarifies how values of higher operations compose, insert, and interact with primary and Whitehead-type structures, and it extends Hilton-type decompositions to a higher-operational level. The paper provides explicit constructions and representatives for Whitehead products (including higher order and rational/integral aspects), and extends these ideas to Lie-Massey products via DG Lie algebra models. Applications include rational models for complex projective spaces and a structured approach to understanding how unstable homotopy groups can be generated by higher operations, with potential for integral refinements and broader ∞-categorical contexts.

Abstract

We explain how the simplicial higher-order unstable homotopy operations defined in [BBS2] may be composed and inserted one in another, thus forming a coherent if complicated algebraic structure.

The algebra of higher homotopy operations

TL;DR

This work develops a coherent algebraic framework for simplicial higher order unstable homotopy operations, formulated in a model-free ∞-categorical setting, and connects these operations to spectral sequence indeterminacies. It clarifies how values of higher operations compose, insert, and interact with primary and Whitehead-type structures, and it extends Hilton-type decompositions to a higher-operational level. The paper provides explicit constructions and representatives for Whitehead products (including higher order and rational/integral aspects), and extends these ideas to Lie-Massey products via DG Lie algebra models. Applications include rational models for complex projective spaces and a structured approach to understanding how unstable homotopy groups can be generated by higher operations, with potential for integral refinements and broader ∞-categorical contexts.

Abstract

We explain how the simplicial higher-order unstable homotopy operations defined in [BBS2] may be composed and inserted one in another, thus forming a coherent if complicated algebraic structure.
Paper Structure (8 sections, 10 theorems, 75 equations)

This paper contains 8 sections, 10 theorems, 75 equations.

Table of Contents

  1. Higher order operations
  2. Simplicial resolutions
  3. Combining higher order homotopy operations
  4. Whitehead products
  5. The algebra of higher homotopy operations
  6. Higher order Whitehead products
  7. Lie-Massey products
  8. Complex projective spaces

Key Result

Proposition 2.10

Let ${\mathbf W}\sb{\bullet}$ be a simplicial space in which each ${\mathbf W}\sb{n}$ is weakly equivalent to a wedge of simply-connected rational spheres. Then the Bousfield-Friedlander spectral sequence for ${\mathbf W}\sb{\bullet}$ collapses at the $E\sp{2}$-term.

Theorems & Definitions (44)

  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.3
  • Definition 2.5
  • Proposition 2.10
  • proof
  • Remark 2.11
  • Corollary 2.12
  • ...and 34 more