Topology Vol. 8, pp. 219-232. Pergamon Press, 1969. Printed in Great Britain zyxwvutsrqponmlkjihgfedcbaZYXWVUTS ON THE SPHERE OF ORIGIN OF INFINITE FAMILIES IN THE HOMOTOPY GROUPS OF SPHERES BRAYTON GRAY (Received 15 July 1968; revised 2 December 1968) THE OBJECT of this paper is to prove some general theorems on the existence of families of elements in the homotopy groups of spheres, and deduce from these a best possible result on the sphere of origin of the elements with nontrivial e-invariant in the p-primary component for p > 2. Recall [l] that e : G, + Q/Z, where G, is the stable r-stem and that G, z (Im J) 0 (ker e), for p > 2. We conjecture that the elements constructed are in the image of the stable J-homomorphism. This is true mod p. Let {A, B} denote the set of stable homotopy classes of maps from A to B. If CC E (A, B4} where B4 is a Moore space of homological dimension q, we write czt for any desuspension of CL to a map CI, : Srm4A + B. If such an c(, exists we will say that a exists on B,. If u E {BP, B4} write /aI = p - q. Throughout this paper, a : Sp + Sq will be a fixed map and we will pick a fixed homotopy H : mcL N 0, where the order of CI divides m. We assume m and lcll = p - q are both odd. THEOREM5.4. With the above assumptions there exist maps z(r) : Sr(iai+l)+q-l + Sq such that (1) ma(r) N 0 (2) cl(l) = a (3) (I + sMr>2,44 = 0 (4) 4r + s& E I@),, ml, c@)> (5) r4r + s)2q E {(r + s)4r)2q, 4s), ml> THEOREM 5.8, With the above assumptions, suppose c(~+~ = n/3 where n is odd. Choose a homotopy R : rnnfl N 0 and define j?(s) as above. Then there exists y : St--* S2q+k such that my = P(m)2q+k, where t = m(lal + 1) + 2q + k - 1. Furthermore, fy belongs to the Toda bracket: (alq+k 9 (:i,)y (mm 4, B(m - 2) ) THEOREM 6.1. In addition to the above assumptions assume q is odd. zyxwvutsrqponmlkjihgfedc If m E 3(mod 9) assume (S2c()a, = 0 where GIN E n,+,(P+‘) is any element of order 3. Then there exist 219