On Adding Relations to Homotopy Groups

In a recent paper2 I described in algebraical terms the relation between 7rn-i(X) and 7rn1-(X*), and also the relation between 7rn(X) and 7rn(X*) in case each of fi(St'') is homotopic to a point. Here we study the relation between 7rn(X) and 7r,,(X*) when the maps fi(S'-') are arbitrary. There is a considerable difference between the cases n = 2 and n > 2. In case n > 2 the relation between 7rn(X) and 7rn(X*) is expressed in terms of a product a A e 7rm+n-1(X), where a e rm(X), 0 E 7rn(X). The case n = 2 is, in many ways, the more interesting of the two. Among other things a method is found for calculating3 7r2(K) algebraically, where K is any simplicial complex. Of course K. Reidemeister's4 theory of homology in K, the universal covering complex of K, together with a theorem due to, W. Hurewicz5 lead to a theoretical definition of 7r2(K), which may be stated in purely algebraic terms. But since there is no general algorithm for deciding whether or no given elements Pi X n * p*. in the group ring, 9?, of K, satisfy given equations