Date: Thu, 29 Apr 1999 14:30:57 CDT From: "Martin C. Tangora (312) 996-3064" Subject: Re: (job and) question > Date: Thu, 29 Apr 1999 13:10:02 -0400 (EDT) > From: Jie Wu > Subject: Size of homotopy groups > > The following questions were asked by Wolfgang Ziller: > > Are there any descriptions about the size (order) t(k) of \pi_k(S^n)? > > Is t(k) bounded by an exponential function on k, a polynomial function on > k or even better? We had this thread in January 1999. At the time I was going to post the following references but was confused about one point, wrote to Doug Ravenel about it, and never got back to the list. Now I can't find Doug's reply. Must be getting old. Anyway, Hans-Werner Henn has a paper on the growth of homotopy groups and he shows that it's exponential. I proved that the same growth rate holds for the E_1 term of the Adams spectral sequence, by which I mean the lambda algebra; so it seems most plausible that the E_2 term obeys the same law. Here's the draft note of February 2, 1999: = = = = = Some references that don't seem to be widely known: 1. A long version of Anick's work, "The computation of rational homotopy groups is #P-hard," is in the proceedings of the 1986 conference at UIC on "Computers in geometry and topology." That's the title of the volume, edited by me, published by Marcel Dekker, No. 114 in Dekker's Lecture notes in pure and applied math., appeared 1989. Unfortunately the acquisition editor left Dekker, and the new people gave the book a price that was tantamount to burning the entire press run. For a couple of years it was a relative bargain at the annual meetings, but now it is hard to find. Same volume has Milnor's paper on self-similarity & hairiness in the Mandelbrot set. 2. More to the point, it has Ravenel's "Homotopy groups of spheres on a small computer" (I believe it was a Commodore 64) which contains a section "Empirical observations on the growth of homotopy groups" where the growth rate is discussed. 3. Trying to take a baby step toward the complexity question in my Memoir (Mem AMS 337 1985), I did an empirical computation of the growth of the E2 term for p=2. But Hans-Werner Henn did a paper on the growth rate of homotopy groups, in Manuscripta Math. 56 (1986). Both of us found the same growth rate! I have suggested (Europ. J. Combin. 1991 p.441) that this is because in both cases an EHP-type sequence is the basis for the computation. The common growth rate is exponential with a factor (or base) between 1 and 2, depending on which prime and on how you are measuring degree. Professor Martin C. Tangora Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago, Mail Code 249 851 South Morgan Street, Room 322 Chicago, Illinois 60607-7045 tangora@uic.edu