157 158 R. JAMES MILGRAM Ext 1 .) Similarly, the mod p operations can be calculated. We have ([12]) and a Q β^p i{v~ι) {hi) Φ 0 for i ^ 0. Thus we recover the result of Liulevicius and Shimada on the elements of mod p Hopf invariant one. In §6, we calculate the complete action of the Sq* operations in Έxt s J {2) (Z 2 , Z 2 ) for t -s < 42. It turns out that routine relations among the various classes, together with the differentials of 4.1.1 (a), determine all 9 2 differentials in this range.

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... I am indebted to M. Tangora for showing me how to obtain some of the more obscure 3 2 differentials in this way.) One surprising result is that 3 2 (c 2 ) = h o f l9 a differential in the 41 stem which was overlooked in [13]. This differential, in turn, implies vθ 4 Φ 0 (which contradicts a result in [13] ) where <9 4 is the class corresponding to the Kervaire invariant 1 manifold in dimension 30. This in turn forces a 3 3 differential in the 34 stem. Once these two differentials are accounted for, there seem to be no further corrections necessary in [13] , and we can thus assume the two primary components of π s t (S°) known for t < 44. For further details, see [5] . Perhaps equally surprising, the technique used to prove Theorem 4.1.1 is purely geometric in nature. We never need mention secondary cohomology operations or even primary ones. Next, we study some of the ways in which the operations imply higher order differentials. Here the answers are not as satisfactory as before. However, we do succeed in characterizing all primary differentials through 9 20 on these elements. More exactly, Theorem 5.1.1 characterizes the first possible nonzero d fc (S<f (α)) for k < 20, provided d ά (a) = 0 for j ^ k + 1. However, there are no places in the first 40 stems where such differentials occur (except, indirectly, the differential 3 3 (r) = h^l), so in the absence of examples, it seemed fruitless to pursue the matter further. Also, there are further applications of these geometric techniques. For example, in §7 we give a very direct proof that θ 4 is nonzero. Moreover, by using similar techniques with the two-cell-complex S 15 U 2 e 16 , together with the fact that (<9 3 ) 2 = (σ) 4 = 0, one obtains a proof that θ δ is nonzero. Similarly, using the result of Barratt-Mahowald that (β 4 ) 2 = 0, one obtains the existence of θ 6 (see [22] for details). Finally, I would like to take this opportunity to express my thanks to D. S. Kahn for sharing his insights with me, and the Centro de Investigation for their support while this research was carried out.