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
Famous quotes containing the word calculations:
“The vulgar call good fortune that which really is produced by the calculations of genius.”
—Ralph Waldo Emerson (1803–1882)
“Now, since our condition accommodates things to itself, and transforms them according to itself, we no longer know things in their reality; for nothing comes to us that is not altered and falsified by our Senses. When the compass, the square, and the rule are untrue, all the calculations drawn from them, all the buildings erected by their measure, are of necessity also defective and out of plumb. The uncertainty of our senses renders uncertain everything that they produce.”
—Michel de Montaigne (1533–1592)
“What, really, is wanted from a neighborhood? Convenience, certainly, an absence of major aggravation, to be sure. But perhaps most of all, ideally, what is wanted is a comfortable background, a breathing space of intermission between the intensities of private life and the calculations of public life.”
—Joseph Epstein (b. 1937)