Three postings related to fundamental groups.
a. A short one by Rudyak
b. Comments by Witbooi on Goodwillie's posting earlier today.
I have left in Goodwillie's message for comparison
c. A short TeXed document by Jack Morava......DMD
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Subject: Re: Goodwillie on compactification
Date: Mon, 13 Oct 2003 11:38:26 -0400 (EDT)
From: Yuli Rudyak
Tom Goodwillie wrote:
>8. The colleague of Rudyak who first brought this up says that he can
>rule such things out in the case when X is paracompact. So I gather
>that his assertion is, as I thought, that a connected CW complex is a
>path component of its Stone-Cech compactification.
Really, my colleague James Keesling sketched me the proof of this claim (that a
connected CW space is a path component of its Stone-Cech compactification) .
So, every group is the fundamental group of a compact Hausdorff space.
Keesling's proof is non-trivial (although not very long), and I think it makes
sense to write it down and put in an archive. So, hopefully, we will have a
possibility to see it soon.
Yuli
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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.
B. Regarding the latter part of item 7, consider the following example.
Let X=(-1,1) be the open interval and let Y=[-1,1] be the closed interval
Suppose that p:I->BX is a path such that p(t) is only in X when t is 0.
By the universal property of the Stone-Cech compactification,
for the inclusion map f: X \subset I, and since Y is compact Hausdorff,
there exists a map q:BX \to Y such that the composition of the natural map
X \to BX followed by q coincides with f. But then the image J of the path
q \circ p (i.e. p followed by q) consists of $p(0)$ together with at least
one of the points in {-1,1}, i.e., J consists of at least two and at most 3
points and therefore cannot be connected.
Peter Witbooi
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>>> Don Davis 10/13/03 01:46PM >>>
Subject: Stone-Cech compactifications
Date: Sun, 12 Oct 2003 22:46:59 -0400
From: Tom Goodwillie
Hi everybody. I've embarrassed myself again by shooting my mouth off.
I was so happy to have learned a few cool facts about point set
topology that I went and overstated some things. I shouldn't have
been so hasty to show off my new-found knowledge.
I made two mistakes. First, to argue that sequences cannot escape
from X in BX I should have assumed X was normal. Second, and more
importantly, it's not clear that paths cannot escape just because
sequences cannot escape.
Let me share a few things that I have learned. Probably everybody
should know most of this this stuff, but I didn't, and I know I'm not
the only topologist who knows very little about general topology, so
I figure somebody might get something out of this.
1. For any space X there is a space called BX (well, it should be
beta but B is easier to type), defined as a subspace of the product
of a lot of copies of I, one for each continuous map from X to I. BX
is the closure of the image of the obvious map from X to this product.
2. BX is compact Hausdorff, of course, and B is a functor from spaces
to compact Hausdorff spaces. It is left adjoint to the forgetful (or
inclusion) functor.
3. The canonical map from X to BX is an embedding (i.e. a
homeomorphism followed by the inclusion of a subspace) if and only if
X is completely regular (i.e. if in X points are closed and points
can be separated from closed sets by real functions).
In this case (at least) BX is called the Stone-Cech compactification of
X.
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.
5. If X is regular then for any non-convergent sequence in X there
are disjoint closed sets A and B in X each of which contains some
subsequence. (Case 1: The sequence has a limit point x. Then since
the sequence does not converge to x, there is an open nbhd X-A of X
whose complement A contains a subsequence. Since X is regular, X-A
contains a closed nbhd B of X. Since x is a limit point, B contains a
subsequence. Case 2: There is no limit point. Then the set of points
in the sequence is closed, discrete, and infinite, so write as the
union of two infinte sets A and B ...)
6. If X is normal then, using a function f:X->I that equals 0 on A
and 1 on B, we conclude from 5 that a sequence in X cannot converge
in BX except by converging in X.
7. I wanted to conclude from 6 that a path in BX cannot be partly in
X and partly out of X. But I was foolishly imagining that X was open
in BX. Silly me! That would make X an open subset of a compact
Hausdorff space, therefore locally compact. I don't know why you
can't have a path p:I->BX such that p(t) is only in X when t is 0. On
the other hand, I can't think of an example.
8. The colleague of Rudyak who first brought this up says that he can
rule such things out in the case when X is paracompact. So I gather
that his assertion is, as I thought, that a connected CW complex is a
path component of its Stone-Cech compactification.
Tom Goodwillie
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Subject: Ideleclass
Date: Mon, 13 Oct 2003 07:59:06 -0400 (EDT)
From: Jack Morava
The document below was motivated by Haynes' note, about
when (if ever) the fund group can be a rational vector space. There
seem to be natural examples of such things (classifying space of
the solenoid, unless I misunderstand something.
There do seem to be interesting connections with number theory, but
the literature there is scattered, which is why I put in a bunch
of references.
thanks,
Jack
\documentclass{letter}
\usepackage{amsfonts}
\begin{document}
\newcommand{\A}{{\mathbb A}}
\newcommand{\PP}{{\mathbb P}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\T}{{\mathbb T}}
\newcommand{\Z}{{\mathbb Z}}
Dear Don,
I believe that some questions about classifying spaces of compact
topological abelian groups are not very well-understood, and I
bring this up because their properties may be relevant to recent
postings about fundamental groups. The solenoid group (which can be
defined in various ways, for example as the inverse limit of the family of
coverings of the circle, ordered by divisibility) is maybe the most
interesting example. (From another point of view, it the Bohr
(almost-periodic) compactification of the real line.) Joel Cohen
[Proc. London Math. Soc. 27 (1973)] showed that if $A$ is a compact
abelian topological group, then the homotopy groups of its classifying
space $BA$ can be calculated as
\[
\pi_*(BA) \cong {\rm Ext}^{2-*}(\check A,\Z)
\]
where $\check A = {\rm Hom}_c(A,\T)$ is the Pontrjagin dual of $A$. In
particular, this implies that if $A$ is the Pontrjagin dual of $\Q$,
then its classifying space is an Eilenberg-MacLane space of type
$K({\rm Ext}(\Q,\Z),1)$. This seems remarkable to me: it says that the
group $\check \Q$ [which is known to be connected, as a topological
group - cf. the exercises in Eilenberg \& Steenrod] is not path-connected,
but rather has something like $\A/\Q$ (where $\A = \hat \Z \otimes \Q$ is
the ring of finite idel\'es) as its group of path-components. I guess that's
possible, but it's unsettling; maybe I've got something wrong here.
Such groups actually arise in `Nature': a number field $K$ [ie a finite
extension of $\Q$] has an interesting associated idel\'e-class group $C_K$
[cf. eg. Cassels \& Fr\"ohlich II \S 17], whose connected component $C^0_K$
is most easily described in terms of its Pontrjagin dual
\[
\check C^0_K \cong \R \times \Z^c \times \Q^{c+r-1} \;,
\]
where $c$ is the number of complex Archimedean primes of $K$, and $r$ is the
number of real ones. [This is in the old Artin-Tate seminar notes [Ch. 9]; a
more modern reference is Ch. VIII \S 2 of Neukirch, Schmidt, and Wingberg,
Cohomology of number fields, Springer Grundlehren v. 323. It's
related to what Deligne [in his paper on $\PP_1 - \{0,1,\infty\}$, cf.
\S 1.16]] calls the `Betti cohomology' of the number field.]
I became interested in these gadgets after some conversations with Sasha
Goncharov at Brown, a couple of years ago. Weil studied certain extensions
\[
1 \to C^0_L \to W(L/K) \to {\rm Gal}(L_{\rm ab}/K) \to 1 \;,
\]
for $L/K$ Galois, with $L_{\rm ab}$ the maximal abelian extension of $L$,
cf. Tate in Cassels \& Fr\"ohlich [VII \S 11]; the middle term of the
extension is sometimes called the Weil group [cf. Artin-Tate Ch. 15].
Recently Connes [Sym\'etries Galoisiennes et renormalisation, {\tt
math.QA/021119} [p. 15]] has drawn attention to the interest of these groups,
and I compiled these notes in hopes of someday having a shot at the Postnikov
tower of $BW(L/K)$; corollary 8.2.6 of Neukirch et al. might be relevant
\dots
Hope this makes sense to somebody! All the best,
Jack
\end{document}
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