TITLE: The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres
QUESTION [13 upvotes]: Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of spheres?

REPLY [3 votes]: I think it is important to remember that the Brown-Loday theorem concerns colimits of cat-n groups obtained from an n stage filtration of the underlying space. Moreover, cat-n groups can only provide information on the n-type.
So, if you wanted to compute pi_200 of the two sphere, you would need a cat-200 group (at least). And if you wanted to apply the Brown_Loday VK theorem you would need a 200 stage topological filtration of the two sphere. Any thoughts ? or do I have this wrong?