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Loop space decompositions of moment-angle complexes associated to flag complexes

Lewis Stanton

Abstract

We prove that the loop space of the moment-angle complex associated to the $k$-skeleton of a flag complex belongs to the class $\mathcal{P}$ of spaces homotopy equivalent to a finite type product of spheres and loops on simply connected spheres. To do this, a general result showing $\mathcal{P}$ is closed under retracts is proved.

Loop space decompositions of moment-angle complexes associated to flag complexes

Abstract

We prove that the loop space of the moment-angle complex associated to the -skeleton of a flag complex belongs to the class of spaces homotopy equivalent to a finite type product of spheres and loops on simply connected spheres. To do this, a general result showing is closed under retracts is proved.
Paper Structure (14 sections, 37 theorems, 61 equations)

This paper contains 14 sections, 37 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminary Material
  3. Idempotent Matrices
  4. Atomicity of loops on spheres
  5. James-Hopf maps and Hopf invariants
  6. Hurewicz images
  7. Preliminary loop space decompositions
  8. Closure of $\mathcal{P}$ under retracts
  9. Setup
  10. Case 1
  11. Case 2
  12. Case 3
  13. Conclusion of proof
  14. Loop space decompositions of pushouts of polyhedral products as a product of spheres and loops on spheres

Key Result

Theorem 1.1

Let $k \geq 0$, and let $K$ be the $k$-skeleton of a flag complex on the vertex set $[m]$ and $A_1,\cdots,A_m$ be path connected $CW$-complexes such that $\Sigma A_i \in \mathcal{W}$ for all $i$. Then $\Omega (\underline{CA},\underline{A})^{K} \in \mathcal{P}$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • ...and 52 more