The Dual Polyhedral Product, Cocategory and Nilpotence

TL;DR

This paper introduces the dual polyhedral product concept, linking cocategory and nilpotence, and applies it to solve a longstanding question about the homotopy nilpotency class of certain spaces.

Contribution

It defines the dual polyhedral product and establishes its properties, connecting cocategory with homotopy nilpotency, and applies these results to complex Lie groups and p-compact groups.

Findings

Weak cocategory equals homotopy nilpotency class for simply-connected spaces

Solved Ganea's fifty-year-old question on cocategory and nilpotence

Determined nilpotency classes for exceptional Lie groups and p-compact groups

Abstract

The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik-Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if X is a simply-connected space then the weak cocategory of X equals the homotopy nilpotency class of the based loops on X. This answers a fifty year old question posed by Ganea. The methods are applied to determine the homotopy nilpotency class of quasi-p-regular exceptional Lie groups and sporadic p-compact groups.