  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.
