HOMOTOPY PULL BACKS, HOMOTOPY PUSH OUTS AND JOINS

We show here how the techniques based on homotopy pull backs and push outs lead to simple proofs for apparently difficult (known or unknown) results. They can be used not only in the category of topological spa ces, but also in any Quillen's model category. Many of them rely on th e two 'join theorems' we prove here. Further applications are the stud y of holonomy, or of the Lusternik-Schnirelmann category.