Subject: Re: Kuperberg and Stasheff Date: Sun, 24 Jan 1999 15:18:44 -0500 From: Clarence Wilkerson >> State of the art for algorithms for say homotopy groups of classifying spaces? Classifying spaces is a broad term. If one interprets it as BG for G a connected Lie group, one knows 1) \pi_k(BG) iso to \pi_{k-1}(G) 2) For any odd prime p and G simply connected, G localized at p is h.e. to a product of odd spheres if ( Serre ) and only if ( Wilkerson ) 3 + 2(p-1) > degree (top indecomposable in H^*(G,\QQ)) so 3) for a given G, for most primes, calculating \pi_{k-1}(G) is equivalent to the problem of calculating the homotopy groups of spheres. For the smaller primes, there are various fibrations linking the problems. On the other hand, if G is not connected, but, say discrete, and one looks at BG completed at some prime, then there is no easy connection between the homotopy groups and group theoretic properties of G. Fred Cohen and Ran Levi have studied this. An early exampe due to Sullivan is the split extension 0 --> Z/p^{\infty}Z --> G --> Z/2Z --> 0, where the action is by multiplication by -1 on the kernel. Then for odd p, the completion of BG is the same as for the completion of BSU(2), and hence the homotopy groups are basically those of S^3=SU(2) Finally, besides the study of periodic phenomena in the stable case, one has in the unstable case the work of Selick, Cohen-Moore-Neisendorf bounding the"exponent" of the homotopy groups of spheres. For example, for odd p, the p-primary components of the higher homotopy groups of the 3 sphere are all annihilated by multiplication by p. Surely one test of the value of a problem is in the mathematics generated by efforts to solve it. For the case of the calculation of homotopy groups of spheres, both stable and unstable, there is large interesting and varied body of work attacking it. Clarence Wilkerson (Just do it! )