The homotopy type of the polyhedral product for shifted complexes

TL;DR

This paper proves a conjecture relating the homotopy type of polyhedral products for shifted complexes to wedges of suspensions of smash products, generalizing previous results and connecting to multiple mathematical fields.

Contribution

It establishes the homotopy equivalence of polyhedral products for shifted complexes to wedges of suspensions of smash products, extending prior work to a broader class of spaces.

Findings

Polyhedral product for shifted complexes is homotopy equivalent to a wedge of suspensions of smash products.

Generalizes earlier results for loop spaces to broader spaces.

Connects homotopy theory with combinatorics and toric topology.

Abstract

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices, X_1,..., X_n are spaces and CX_i is the cone on X_i, then the polyhedral product determined by K and the pairs (CX_i,X_i) is homotopy equivalent to a wedge of suspensions of smashes of the X_i's. This generalises earlier work of the two authors in the special case where each X_i is a loop space. Connections are made to toric topology, combinatorics, and classical homotopy theory.