Iterated Higher Whitehead products in topology of moment-angle complexes

TL;DR

This paper investigates the topological structure of moment-angle complexes, showing that not all wedge sphere complexes are realizable by iterated higher Whitehead products, but the class of such realizable complexes is extensive and well-structured.

Contribution

It provides a counterexample to the conjecture that all wedge sphere complexes are realizable by Whitehead products and establishes the structural properties of the realizable class.

Findings

Counterexample of a wedge sphere complex not realizable by Whitehead products

The class of Whitehead product realizable complexes is closed under specific operations

Existence of a minimal complex realizing a given iterated Whitehead product

Abstract

In this paper we study the topological structure of moment-angle complexes $\mathcal{Z_K}$. We consider two classes of simplicial complexes. The first class $B_{\Delta}$ consists of simplicial complexes $\mathcal{K}$ for which $\mathcal{Z_K}$ is homotopy equivalent to a wedge spheres. The second class $W_{\Delta}$ consists of $\mathcal{K}\in B_{\Delta}$ such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that $B_{\Delta} = W_{\Delta}$. In this paper we show that this is not the case. Namely, we give an example of a simplicial complex whose corresponding moment-angle complex is homotopy equivalent to a wedge of spheres, but there is a sphere which cannot be realized by any linear combination of iterated higher Whitehead products. On the other hand we show that class $W_{\Delta}$ is large enough. Namely, we show that the class $W_{\Delta}$ is closed with respect to two explicitly defined operations on simplicial complexes. Then using these operations we prove that there exists a simplicial complex that realizes any given iterated higher Whitehead product. Also we describe the smallest simplicial complex that realizes an iterated product with only two pairs of nested brackets.