Three more postings on pi_1............DMD __________________________________________________________ Subject: pi_1 and H_1 Date: Mon, 13 Oct 2003 18:26:54 -0400 From: "Claude Schochet" This discussion seems to be coming my way, so I thought that I'd throw in a comment or two. >From the point of view of functional analysis and index theory, the more natural question would be: A) Which groups appear as the fundamental group of (separable) compact metric spaces? I have no idea about that one, but suppose we look at the image of the fundamental group in H_1. Which version of H_1 ? Well, I prefer Steenrod homology (dual to Cech cohomology). So we could ask: B) Which abelian groups appear as H_1 (Steenrod) of (separable) compact metric spaces? Now one can say quite a bit about this, and it links up to Jack's solenoid comment. Steenrod invented Steenrod homology (in 1940) to fix the fact that Cech homology wasn't a homology theory. The group maps onto Cech homology (which typically will be profinite). It misses being 1-1 by a lim^1 which, in the case of a solenoid, Steenrod computed. (This was before he and Eilenberg introduced the word "functor" not to speak of "derived functor"). The point is that there is a nice lim^1 term sitting as the closure of zero in H_1, so we reach an algebraic question: C) Which abelian groups can appear as lim^1 of an inverse sequence of finitely generated abelian groups? Now quite a bit is known about that question. For example, Warfield proved that such a group must be cotorsion [an abelian group is cotorsion if Ext(Q,G) = 0 ]. Brayton Gray proved that the group must be zero or uncountable. lim^1 is related to Pext (Pext is to pure exact sequences what Ext is to exact sequences). I gathered up quite of bit of information, including a lot of examples, in my Pext Primer http://nyjm.albany.edu:8000/m/2003/1nf.htm Claude ____________________________________________________________ Subject: Re: 3 0n pi_1 Date: Mon, 13 Oct 2003 18:49:52 -0500 From: "Martin C. Tangora" I would be grateful if Mr Witbooi would clarify what is "obviously false" in Professor Goodwillie's point 4. In fact I have been teaching that equivalence in my point-set topology courses for almost 40 years. I hope he is not just going to say "because you forgot to say Hausdorff." If you don't require your completely regular spaces to be T1, then you would have a problem. Being rusty is no excuse for saying that things are obviously false when they are actually well known theorems. Or what am I missing? At 03:56 PM 10/13/2003 -0400, you wrote: >Three postings related to fundamental groups. >a. ... >b. Comments by Witbooi on Goodwillie's posting earlier today. > I have left in Goodwillie's message for comparison >c. ... .....DMD >__________________________________________________________ > >Subject: Re: Goodwillie on compactification >Date: Mon, 13 Oct 2003 18:35:47 +0200 >From: "Peter Witbooi" > > I have two remarks on Goodwillie's message below. I do not want >say more than this because my General Topology is also a bit rusty. > >A. The statement 4 below, on those "equivalences", is obviously false. >... > >Peter Witbooi > >>>> Don Davis 10/13/03 01:46PM >>> >Subject: Stone-Cech compactifications >Date: Sun, 12 Oct 2003 22:46:59 -0400 >From: Tom Goodwillie > >... >4. It follows from 3 that a implies d below, and therefore a, b, c, d >are equivalent > >a. X is completely regular >b. X is a subspace of a completely regular space >c. X is a subspace of a normal space >d. X is a subspace of a compact Hausdorff space > >Note that regularity and complete regularity are (obviously) >inherited by subspaces, but normality is not. >... > >Tom Goodwillie Martin C. Tangora University of Illinois at Chicago tangora@uic.edu ___________________________________________________________ Subject: Re: 3 0n pi_1 Date: Tue, 14 Oct 2003 08:46:15 +0200 From: "Peter Witbooi" As I was on my way to the office this morning I realized that my statement A of the previous message below is wrong and I wish to retract it, pardon. Peter.