Rational Functions of a Complex Variable and Related Potential Curves.

This implies that in this case n2r is the kernel of the natural homomorphism of 7r2 into the 2-dimensional integral homology group of X. Comparing this with recent results of Hopf5 yields numerous examples of spaces with k3 $ 0. 5. Generalizations.-1°: The construction of k3 generalizes in an obvious fashion to yield a cohomology class kn+1 e Hn+l(irj, 7r,,), provided that 7ri = 0 for 1 < i < n. Theorems 1 and 2 generalize accordingly. 20: Assuming 7r3 = , the groups H3(X, G) may be expressed by means of 7ri, 7r2 and k3. The algebraic constructions involved are quite cumbersome. 30: If ir, acts as a group of operators on G, and H2(X, G) denotes the cohomology group of X with G as local coefficients,6 Theorem 2 is still valid provided that (3.4) is replaced by g2(wa) = wg2(a). 40: Instead of working in the space X itself, as above, one could use the universal covering space X of X and regard 7r1 as a group of operators on X. In this approach homotopy arguments are replaced by homology arguments, and some additional generality is gained. This approach will be used in the full exposition elsewhere of the results.