Publisher Summary The theory of obstructions to extensions of mappings has simplified and unified many results in homotopy theory. This chapter presents a dual theory in which the problem is to deform a map ( X , A ) →( Y , B ) into one with values in a subspace Y ' ( B ⊂ Y ') by a homotopy of the form ( X , A ) → ( Y , B ). This is referred to as problem of compressing f into Y '. When the spaces considered are suitably restricted, this leads to obstructions that are cycles or homology classes in ( Y , B ) with cohomotopy groups of ( X, A ) as coefficients. The chapter describes such obstructions.