Three more postings on pi_1............DMD
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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
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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
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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.