Homotopy Classification of Maps of the Sphere

Abstract

In many problems of mathematics and physics the question arises of classifying maps from the k-sphere S ^k into some space E. Recall that two maps S ^k → E are called homotopic if they can be joined by a continuous family of maps S ^k → E; we also say that one map can be deformed into the other, and that they belong to the same homotopy class. In this chapter we study the set {S ^k, E} of homotopy classes of maps S ^k → E. In Chapters 1 and 2 we solved this problem in the cases E = R ^m and E = S ^r, for r ≥ k. We established that {S ^k, R ^m} for all m and {S ^k, S ^r} for r > k have a single element, and that {S ^k, S ^k} is in one-to-one correspondence with the integers, the correspondence being given by the degree of the map.