Calculations
The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.
Adams' original use for his spectral sequence was the first proof of the Hopf invariant 1 problem: admits a division algebra structure only for n = 1, 2, 4, or 8. He subsequently found a much shorter proof using cohomology operations in K-theory.
The Thom isomorphism theorem relates differential topology to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, Milnor and Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism. Further, Milnor and Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented cobordism ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel–Whitney numbers agree.
Read more about this topic: Adams Spectral Sequence
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