On the Stability of Systems of Differential Equations.

H(a) is h times a generator of wr2,.(S2t1-). Thus Hmay be regarded as.a generalization of. the Hopf invariant. Let R,1 be the rotation group of SI-', 7r the mapping of RT_i into SIwhich sends each rotation r into the image under r of a fixed point yo e St-'. If p < 2r 2 andf is a mapping of SP-1 into R, 1, thenf defines a mapping F: SPX SI -I SI-' of type (7ra, 1), where a is the element of 7rp.(R,_) represented byfand 1 is the elementof 7rr(SI,) represented by the identity map. Then F determines an element y of 'rp+ (S) as in the preceding paragraph, with H(Q) = the r-fold Einhangung Eira of ira. Since p < 2r 2, E is an isomorphism2 and it follows that if ira 0 0, then y d 0. This result can be used to construct essential maps of S' into Sr with n = 12, 14, 8k and 16k + 2, and r = 6, 7, 4k and 8k, respectively. If r = 2, 4, 8, Hurewicz and Steenrod5 have proved that 7r.(Sr) is isomorphic with the direct sum _r(Sir-l) + irn_(SI-'). This isomorphism determines a homomorphism H' of ir(S'r) into w"(S2r-'). It is then easy to see that if n < 3r 3, then H' = H. 3. Let X be an arewise connected space, f a map of S5 into Sr, and g a map of Sr into X. The correspondence (f, g) gf defines an operation associating with a e rn(S), 13 e ir,(X) an element (3.a e ir,(X). It is known that the left distributive law ,B(a, + a2) = ,Bal + , a2 holds, while the corresponding right distributive law is in general false. Using the homomorphismH defined above, we can prove: if n < 3r 3, then