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8:28 AM
@EricPeterson I'm confused. Doesn't this imply that this this spectral sequence is pointiwise degenerate at a finite page (in the sense that for every coordinate on the plane there's there's a page s.t. after it all the differentials in and out are trivial).
I just read the classical proof that the sphere is Harmonic and this was one of the steps but I wasn't sure it implied what I thought it implied.
As I understand the argument is as follows:
The MU-Adams Novikov filtration on the homotppy groups of the fiber on C_n(S)--->S---->L_nS is finite. And the maps C_n(S)--->C_{n-1}(S) are zero on MU-homology and therefore lower the filtration. This implies that after taking pi_k of the chromatic tower we get an eventually constant pro-system of abelian groups.
Then I think this implies that at every column (adams grading) this spectral sequence degenerates at a finite page. Does that sound right?
If this is true then something really interesting is happening in the chromatic tower. In every topological degree the height n monochromatic sphere only talks to monochromatic pieces of neighbouring heights (the distance can grow but its only a finite number of neighbours). It would be interesting to know a bound for this distance, which somehow measures how complicated is the gluing between the monochromatic pieces of the sphere...
 
 
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