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 ΩX. This answers a fifty year old problem 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.
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