ABSTRACT: Some of the important achievements in homotopy theory during the
 last decade are related to the nilpotence theorems of Nishida and Devinatz-Hopkins-Smith,
 and their unstable analogues due to Bousfield and Dror Farjoun.  These theorems grew out
of work done by, among others, Mahowald, Miller,  Ravenel, Wilson . The aim of this  talk is
to describe and explain a connection between the nilpotence theorems and two geometric relations.
These relations are associated to two ways of building spaces out of a chosen space. I will present
several examples illustrating typical phenomena.