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Sep 24, 2010 at 21:24 comment added David Roberts I was going to add: but with cat^n groups you can't just calculate pi_n of anything straight out, but get the whole Pi_n first, then restrict, but Tim beat me to it. :)
Sep 24, 2010 at 18:05 comment added Tim Porter By the way, the beauty of Loday's model is that a cat^n group is NOT a complicated algebraic object in any essential way. It is a group together with 2n endomorphisms satisfying some simple rules. Cat^n groups are equivalent to crossed n-cubes and they are like crossed modules except in n-directions!
Sep 24, 2010 at 18:02 comment added Tim Porter @Tom My own take on it might be a bit different. There is a curiosity in knowing the homotopy groups of spheres, but that is partially a curiosity in studying the way in which the algebra reflects homotopy structure, so modelling the homotopy 200-type of $S^2$ is perhaps in the long run more central to the understanding of the total homotopy type of $S^2$, than 'merely' knowing the value of $\pi_{200}(S^2)$. The $cat^n$ group ($cat^{200}$-group of $S^2$ would contain an awesome amount of information on certain homotopy operations. I cannot start to think what they might be!
Sep 24, 2010 at 13:41 comment added Tom Goodwillie I suppose that the theorem in question turns the computation of, say, $\pi_{200}(S^2)$ into mere algebra: it tells you how to write down generators and relations for some complicated algebraic object called a cat_n group and then you just have to learn how to extract your answer from that.
Sep 24, 2010 at 8:33 comment added Tim Porter @ Harry I will do but it will be a bit later on as I have to think about how best to answer e.g. to look up one of RB and J-LLs examples!
Sep 24, 2010 at 8:28 comment added David Roberts Hmm, good question. The problem I think moves from calculating homotopy groups of spheres to calculating colimits of cat^n groups. I don't think there is a very big literature on cat^n groups (but I note there is a lot on the nonabelian tensor product - RB has collected them on his website) and tools to deal with them.
Sep 24, 2010 at 6:55 comment added Harry Gindi That is, why can't we use the HHvK theorems (the version you're talking about) to compute the homotopy groups of spheres like we compute fundamental groups using the ordinary vK theorem?
Sep 24, 2010 at 6:49 comment added Harry Gindi @Tim: Could you elaborate on that in an answer?
Sep 24, 2010 at 6:43 comment added Tim Porter @David You are forgetting the version that RB proved with Loday. The end result there is a cat^n group and hence a model of the n+1 type of the 'union'/colimit. This leads to the non-Abelian tensor product and that has been a great idea, but they do not relevant to the unstable homotopy groups of spheres at least as far as I know.
Sep 24, 2010 at 4:53 vote accept Harry Gindi
Sep 24, 2010 at 6:47
Sep 24, 2010 at 4:45 history answered David Roberts CC BY-SA 2.5