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Relations among higher Whitehead maps

Jelena Grbic, George Simmons, Matthew Staniforth

Abstract

We define generalised higher Whitehead maps between polyhedral products. By investigating the interplay between the homotopy-theoretic properties of polyhedral products and the combinatorial properties of simplicial complexes, we describe new families of relations among these maps, while recovering and generalising known identities among Whitehead products.

Relations among higher Whitehead maps

Abstract

We define generalised higher Whitehead maps between polyhedral products. By investigating the interplay between the homotopy-theoretic properties of polyhedral products and the combinatorial properties of simplicial complexes, we describe new families of relations among these maps, while recovering and generalising known identities among Whitehead products.
Paper Structure (14 sections, 36 theorems, 186 equations, 7 figures)

This paper contains 14 sections, 36 theorems, 186 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The higher Whitehead map
  3. The nested higher Whitehead map
  4. Folded higher Whitehead maps
  5. Relations
  6. Relations among higher Whitehead maps
  7. Relations among folded higher Whitehead maps
  8. Folds of identity complexes
  9. Folding and substitution
  10. Proof of main theorem
  11. The relative higher Whitehead map
  12. The inclusion map $j$
  13. The case for the higher Whitehead map
  14. Proof of Lemma \ref{['lm:FiLemma']}

Key Result

Proposition 2.3

Let $f_i \colon (CX_i,X_i) \longrightarrow (Y_i,*)$ be maps of pairs for $i = 1,\dots,m$, and for some $j \in \{1,\dots,m\}$, let $f_j' \colon (CX_j,X_j) \longrightarrow (Y_j,*)$. If $X_j$ is a suspension,

Figures (7)

  • Figure 1: The fold $\psi \colon \{4\} \longrightarrow \{1\}$ of a simplicial complex.
  • Figure 2: Construction of the simplicial complex $\mathcal{L}_{\psi}$.
  • Figure 3: Decomposition of the identity complex $\mathcal{K}_{\{\{1\},\{2,3\},\{4\}\}}$.
  • Figure 4: Two decompositions of $C \Sigma \widetilde{X}_i$, and further decompositions of $D_i^1$ and $D_i^-$ for $\widetilde{X}_i = S^0$.
  • Figure 5: Other subspaces of $D_i^1$ and $D_i^-$ for $\widetilde{X}_i = S^0$.
  • ...and 2 more figures

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 80 more