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The Goodwillie calculus of polyhedral products

Guy Boyde, Niall Taggart

Abstract

We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. We do this by analysing the behaviour on calculus of the Denham-Suciu fibre sequence, the Iriye-Kishimoto decomposition of the polyhedral product constructed from a collection of pairs of cones and their bases, and the Hilton-Milnor decomposition. As a corollary we show that the Goodwillie calculus of these polyhedral products converges integrally and diverges in $v_h$-periodic homotopy unless the simplicial complex is a full simplex.

The Goodwillie calculus of polyhedral products

Abstract

We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. We do this by analysing the behaviour on calculus of the Denham-Suciu fibre sequence, the Iriye-Kishimoto decomposition of the polyhedral product constructed from a collection of pairs of cones and their bases, and the Hilton-Milnor decomposition. As a corollary we show that the Goodwillie calculus of these polyhedral products converges integrally and diverges in -periodic homotopy unless the simplicial complex is a full simplex.
Paper Structure (16 sections, 32 theorems, 92 equations)

This paper contains 16 sections, 32 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Goodwillie calculus
  3. Single-variable Goodwillie calculus
  4. Multi-variable Goodwillie calculus
  5. Integral loop space decompositions of the polyhedral product
  6. A split fibre sequence
  7. The decomposition of Iriye-Kishimoto
  8. The Hilton-Milnor theorem
  9. Goodwillie approximations of polyhedral products
  10. Multi-variable approximations
  11. Single-variable approximations
  12. Comparison with the identity functor evaluated on polyhedral products
  13. Convergence
  14. Multi-variable vs. single-variable convergence
  15. Integral convergence of polyhedral products
  16. ...and 1 more sections

Key Result

Theorem 1

For pointed connected spaces $X_1, \dots, X_m$, there is an equivalence where $\mathbb{L}_m$ is a basis for the free Lie algebra on $m$ generators.

Theorems & Definitions (62)

  • Theorem : Brantner-Heuts
  • Theorem A
  • Theorem B
  • Theorem 2.1.1
  • Remark 2.1.2
  • Theorem 2.2.1
  • Remark 2.2.2
  • Remark 2.2.3
  • Remark 2.2.4
  • Lemma 3.1.1
  • ...and 52 more