The Classification of Continuous Mappings of a Complex into a Sphere, Communication I *

Two continuous mappings f and g of a topological space R into an n-dimensional sphere S_n are said to be equivalent, if these mappings are such that one can be transformed into the other by a continuous deformation. Thus all continuous mappings of a space R into the sphere S_n fall into equivalence classes. The determination of these classes is a problem of current interest in topology. It was completely solved by Hopf for the case in which R is an n-dimensional complex K_n . The case R = K_r (r > n) still awaits a solution.