Hopf Invariants and Browder ' S Work on the Kervaire Invariant Problem

In this paper we calculate certain functional differentials in the Adams spectral sequence converging to Wu cobordism whose values may be thought of as Hopf invariants. These results are applied to reobtain Browder's characterization: if q + 1 = 2*, there is a 2q dimensional manifold of Kervaire invariant one if and only if hi survives to EX(S°). Introduction. In 1969 Browder [2] showed that there is a 2q dimensional framed manifold with Kervaire invariant one if and only if q 41 = 2* and A2 survives to £,00(S,°). More precisely in case q + 1 = 2k, if (M2q, F) is a framed manifold whose Thorn invariant is/ E irlq(S°), then Browder showed that the Kervaire invariant, K(M2q, F), is one if and only if/projects to h\ in £¿'2*+'(5°). On the other hand, the present author [3] introduced homotopy invariants Ix and 72 which are defined on certain subgroups of the homotopy groups trJ(X) of a spectrum X and which take their values in the E2 term of an Adams spectral sequence for X. For the mod 2 Adams spectral sequence of S° and an element/ E ir2q(S°), I2(f) is always defined, and if q + 1 = 2k, A2 survives if and only if there is/ G w2*+<_2(S°) so that 72(/) = A2. Thus we can restate Browder's theorem (at least in dimension q + 1 = 2k) as follows. Theorem 1 (Browder). If (A/2**1-2, F) is a framed manifold with Thorn invariant f E 7r2*+i_2(S°), then I2(f) = K(M2k+'-2,F)h2 in E£2k+'(S°). It is the main purpose of this paper to show that the invariant 72 not only serves to give a convenient expression for Browder's theorem but in fact serves as a vehicle for proving it. Our efforts in this direction center on the mod 2 Adams spectral sequence which converges to Wu cobordism. If MO(vq+x/ denotes the Thorn spectrum for Wu (q + l)-cobordism theory, the Wu manifolds (S? x Sq, q) conReceived by the editors August 7, 1975. AMS (MOS) subject classifications (1970). Primary 55H20, 57D15; Secondary 53G20, 57D20.