证明
Proposition 18 ([All later obstruction candidates]). For k≥2 the complete…Proposition 17 ([The first Adams--Novikov differential]). In the Adams--Novikov…观点
The subalgebra map includes a subset of the exterior and polynomial generators in; its initial-page map is therefore injective in every bidegree. Likewise, multiplication by the polynomial A^a is injective in every bidegree on that page. It is induced by a filtered cochain map: t_1 is primitive, and ∑_ i=1 ^ p-1 ( pi)/(p) [t_1^i t_1^ p-i ] is an actual cocycle, obtained by taking the integral divided reduced coproduct and then reducing modulo p . It has leading class A . For the second comparison shift the source cohomological degree, weight, and auxiliary filtration by 2a , pa , and ap(1+M) . Take the filtered mapping cone of either cochain map. The long exact sequence of associated-graded cohomology, together with injectivity in all bidegrees, identifies its initial cohomology page with the cokernel of the specified page map; there is no extra kernel from the next degree. That cokernel is zero in the stated target bidegree. Finite filtration therefore gives zero cone cohomology there. The ordinary cone long exact sequence gives the required surjection. Transport through the stated commuting comparisons preserves that surjectivity.