A Classification of Mappings of the Three-Dimensional Complex into the Two-Dimensional Sphere

Let Κ and L be two complexes. The family f_t , where t is a real number (0 ≤ t ≤ 1), of continuous mappings of the complex Κ into the complex L is called a continuous deformation of mappings of the complex Κ into the complex L if the function f_t(x) (x ∈ K) is a continuous function of the pair of arguments x,t. Two continuous mappings g and h of the complex Κ into the complex L are said to be nomotopic or equivalent if there exists a continuous deformation f_t transforming the mapping g into the mapping h, i.e., such that g=f₀, h = f ₁. In virtue of this criterium of equivalency all continuous mappings of the complex Κ into the complex L fall into classes of equivalent mappings. A classification of mappings from this point of view, i.e., the determination of more or less effective criteria of equivalency, forms one of the fundamental problems of topology.