[Paper Review] Whitehead products in moment-angle complexes

Kouyemon Iriye, Daisuke Kishimoto|arXiv (Cornell University)|Jun 29, 2018

Homotopy and Cohomology in Algebraic Topology9 references4 citations

TL;DR

This paper establishes that the fiber inclusion $χ_K: \mathcal{Z}_K \to DJ_K$ in the homotopy fibration $\mathcal{Z}_K \to DJ_K \to (\mathbb{C}P^\infty)^m$ is identified with a wedge of iterated higher Whitehead products for a broad class of simplicial complexes, including dual shellable and totally fillable complexes. The result generalizes and corrects earlier work by Grbić and Theriault using a novel approach based on polyhedral products and homotopy colimits.

ABSTRACT

In toric topology, to a simplicial complex $K$ with $m$ vertices, one associates two spaces, the moment-angle complex $\mathcal{Z}_K$ and the Davis-Januszkiewicz space $DJ_K$. These spaces are connected by a homotopy fibration $\mathcal{Z}_K o DJ_K o(\mathbb{C}P^\infty)^m$. In this paper, we show that the map $\mathcal{Z}_K o DJ_K$ is identified with a wedge of iterated (higher) Whitehead products for a certain class of simplicial complexes $K$ including dual shellable complexes. We will prove the result in a more general setting of polyhedral products.

Motivation & Objective

• To generalize and correct the result of Grbić and Theriault that the map $\widetilde{w}: \mathcal{Z}_K \to DJ_K$ is a wedge of iterated Whitehead products for certain simplicial complexes.
• To extend this identification to a significantly larger class of simplicial complexes, including dual shellable and totally fillable complexes.
• To provide a rigorous and simpler proof than previous approaches, which were criticized for potential gaps in reasoning involving rational homotopy theory.
• To establish the result in the broader context of polyhedral products, enabling a more systematic homotopical analysis of $\mathcal{Z}_K$ and $DJ_K$.

Proposed method

• The authors use the framework of polyhedral products to generalize the construction of $\mathcal{Z}_K$ and $DJ_K$, allowing for a more flexible and powerful homotopical analysis.
• They introduce the concept of a 'totally fillable' simplicial complex, defined by the existence of a collection of minimal non-faces whose addition makes the geometric realization contractible.
• The key technical tool is a homotopy commutative diagram involving homotopy colimits and the homotopy colimit decomposition of $\mathcal{Z}_K$ via the nerve of a suitable poset.
• The proof relies on induction on the number of vertices using a 'contraction ordering' of subsets, reducing the problem to iterated Whitehead products.
• They apply the naturality of higher Whitehead products and the homotopy equivalence $\epsilon_M^{-1}$ between a homotopy colimit and the moment-angle complex $\mathcal{Z}_M(\underline{X})$.
• The identification of $\widetilde{w}$ as an iterated Whitehead product is established through a sequence of homotopy equivalences and diagram chasing in the category of pointed spaces.

Experimental results

Research questions

• RQ1 Under what conditions on a simplicial complex $K$ is the fiber inclusion $\mathcal{Z}_K \to DJ_K$ homotopic to a wedge of iterated higher Whitehead products?
• RQ2 Can the result of Grbić and Theriault, which applies to directed MF-complexes, be generalized to a larger class of simplicial complexes with a simpler and more rigorous proof?
• RQ3 Is there a homotopical characterization of $\mathcal{Z}_K$ and $DJ_K$ in terms of Whitehead products that holds beyond the case of boundaries of simplices?
• RQ4 How does the notion of 'fillable' or 'totally fillable' complexes relate to the homotopy type of $\mathcal{Z}_K$ and the structure of the map $\widetilde{w}$?
• RQ5 Can the identification of $\widetilde{w}$ as a Whitehead product be extended to the general setting of polyhedral products?

Key findings

• The fiber inclusion $\widetilde{w}: \mathcal{Z}_K \to DJ_K$ is identified with a wedge of iterated higher Whitehead products for any totally fillable simplicial complex $K$, generalizing previous results.
• For a totally fillable complex $K$, the composite $S^{|σ|+|I|-1} \to \mathcal{Z}_K \xrightarrow{\widetilde{w}} DJ_K$ is homotopic to the iterated Whitehead product $[[\cdots[\widetilde{w}_{\sigma}, \widetilde{a}_{i_1}], \cdots], \widetilde{a}_{i_k}]$ up to sign, where $i_1 < \cdots < i_k$ is a contraction ordering of $I - \sigma$.
• The proof relies on a new homotopy colimit decomposition of $\mathcal{Z}_K$ via the nerve of a poset of subsets, which allows for inductive reduction to simpler cases.
• The authors establish a homotopy equivalence $\epsilon_M^{-1}: W_M(\underline{X}) \to \mathcal{Z}_M(\underline{X})$ that respects the structure of iterated Whitehead products.
• The result holds in the general setting of polyhedral products, showing that the identification of $\widetilde{w}$ as a Whitehead product is intrinsic to the homotopy type of the moment-angle complex.
• The paper corrects a potential gap in earlier work by Grbić and Theriault by replacing rational homotopy arguments with strict homotopy equivalences and diagram chasing.