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Loop space decompositions of moment-angle complexes associated to two dimensional simplicial complexes

Lewis Stanton

TL;DR

The work analyzes loop spaces of moment-angle complexes $\mathcal{Z}_K$ attached to 2D simplicial complexes $K$, proving that $\Omega\mathcal{Z}_K$ always lies in the finite-type product class $\prod(\mathcal{P}\cup\mathcal{T})$, where $\mathcal{P}$ consists of spheres and loop spaces $\Omega S^n$ and $\mathcal{T}$ consists of indecomposable torsion spaces. By establishing that key families ($\bigvee(\mathcal{W}\cup\mathcal{M})$ and $\mathcal{P}\cup\mathcal{T}$) are closed under retracts and deriving robust wedge/smash decompositions, the authors obtain both integral and local (p-local) decompositions and connect these to rational Golodness and neighbourliness conditions on subcomplexes $K_I\in C_K$. They show that, after localisation away from finitely many primes, $\Omega\mathcal{Z}_K$ can simplify to a product of spheres and loops on spheres, with implications for the action of the Steenrod algebra at almost all primes and for conjectures of Anick and McGibbon–Wilkerson. The results provide a framework to understand torsion phenomena in $\Omega\mathcal{Z}_K$ and give explicit, combinatorially controlled primes for which a pure sphere/loop-decomposition holds. Overall, the paper advances the understanding of loop-space decompositions in toric-type spaces through retract-closure results, rational and $p$-local analyses, and localisation techniques.

Abstract

We show that the loop space of a moment-angle complex associated to a $2$-dimensional simplicial complex decomposes as a finite type product of spheres, loops on spheres, and certain indecomposable spaces which appear in the loop space decomposition of Moore spaces. We also give conditions on certain subcomplexes under which, localised away from sufficiently many primes, the loop space of a moment-angle complex decomposes as a finite type product of spheres and loops on spheres.

Loop space decompositions of moment-angle complexes associated to two dimensional simplicial complexes

TL;DR

The work analyzes loop spaces of moment-angle complexes attached to 2D simplicial complexes , proving that always lies in the finite-type product class , where consists of spheres and loop spaces and consists of indecomposable torsion spaces. By establishing that key families ( and ) are closed under retracts and deriving robust wedge/smash decompositions, the authors obtain both integral and local (p-local) decompositions and connect these to rational Golodness and neighbourliness conditions on subcomplexes . They show that, after localisation away from finitely many primes, can simplify to a product of spheres and loops on spheres, with implications for the action of the Steenrod algebra at almost all primes and for conjectures of Anick and McGibbon–Wilkerson. The results provide a framework to understand torsion phenomena in and give explicit, combinatorially controlled primes for which a pure sphere/loop-decomposition holds. Overall, the paper advances the understanding of loop-space decompositions in toric-type spaces through retract-closure results, rational and -local analyses, and localisation techniques.

Abstract

We show that the loop space of a moment-angle complex associated to a -dimensional simplicial complex decomposes as a finite type product of spheres, loops on spheres, and certain indecomposable spaces which appear in the loop space decomposition of Moore spaces. We also give conditions on certain subcomplexes under which, localised away from sufficiently many primes, the loop space of a moment-angle complex decomposes as a finite type product of spheres and loops on spheres.
Paper Structure (15 sections, 35 theorems, 41 equations)

This paper contains 15 sections, 35 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Unique decomposition of H-spaces and co-H spaces
  4. Rational and p-local decompositions of moment-angle complexes
  5. Closure of W cup M under retracts
  6. Closure of P cup T under retracts
  7. Special cases
  8. Review of the proof of Theorem \ref{['thm:Pclosedunderretracts']}
  9. Defining phi'
  10. Case 1
  11. Case 2
  12. Conclusion of proof
  13. Preliminary decompositions of Moment-angle complexes
  14. Loop spaces of moment-angle complexes associated to 2-dimensional simplicial complexes
  15. Loop space decompositions of certain moment-angle complexes after localisation

Key Result

Theorem 1.1

Let $K$ be a $2$-dimensional simplicial complex. Then $\Omega \mathcal{Z}_{K} \in \prod (\mathcal{P} \cup \mathcal{T})$.

Theorems & Definitions (51)

  • Theorem 1.1: Theorem \ref{['thm:2dimloop']} in the text
  • Theorem 1.2: Theorem \ref{['thm:localisedprodofspheres']} in the text
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Theorem 2.6
  • ...and 41 more