From: lrudolph@panix.com (Lee Rudolph)
Subject: the Pr\"ufer manifold (was, Re: Definition of topological manifold)
Date: 18 May 1999 09:14:31 -0400
Newsgroups: sci.math

"Norm Dresner" <ndrez@worldnet.att.net> writes:

>Charles H. Giffen <chg4k@virginia.edu> wrote in article
><37405EA0.193D2B6D@virginia.edu>...
>> 
>> There's also the Pr"ufer manifold which is *really* scary.
>
>Is this an "in" joke or can we all share it?

I may be confused, but I think the Pr\"ufer manifold is
constructed as follows.  As a warm-up, consider the 
Borel-Serre bordification of the upper half-plane:
let H = { x + i y : x, y real, y > 0 } be the open
upper half-plane, say with its standard hyperbolic
metric; let K = H union Q union {oo} be H with all 
the "rational cusps" adjoined; for each p/q in K 
(allowing 1/0 to represent oo, pace Richard Carr), 
let L(p/q) be the set of all geodesics in H with
one "ideal endpont" at p/q.  Then L(p/q) is in
natural bijection with R union {oo} setminus {p/q}
(let a geodesic correspond to its other ideal 
endpoint), and there's a reasonably natural way 
to give BS(Q) = H union all the L(p/q)s the structure
of a smooth 2-manifold-with-boundary whose interior
is H and each component of whose boundary is one of
the L(p/q)s topologized (via that bijection) as 
a real line.  Clearly one can now get a 2-manifold
M(Q) by pasting one copy of the closed upper half-plane
(with boundary R: i.e., I'm taking the closure in C,
not in C union {oo}) to each line L(p/q) along R.

It is my impression that the Pr\"ufer manifold
is, as it were, M(R) = BS(R) union one copy of
the closed upper half plane glued to each line
L(x), where you do your best to interpret my
ad hoc notation consistently, and fill in the
details correctly.

If Peter Nyikos is reading sci.math these days,
he can certaily tell us the right answer (supposing
that Charles Giffen doesn't).

BS(Q), by the way, is a very nice manifold whatever
you think of the rest of this story.  For example,
when you let a subgroup of SL(2,Z) act on H in 
the usual way, the action extends naturally to 
BS(Q) so as to be nice on each R at the boundary;
when the subgroup is such that the quotient of
H is a compact-Riemann-surface-with-punctures,
then the quotient of BS(Q) is a bordered Riemann
surface (whose circle boundary components are the
"ideal boundary components" of the punctured 
surface); etc., etc.

Lee Rudolph
==============================================================================

From: "Charles H. Giffen" <chg4k@virginia.edu>
Subject: Re: the Pr\"ufer manifold (was, Re: Definition of topological manifold)
Date: Tue, 18 May 1999 14:23:28 -0400
Newsgroups: sci.math
To: Lee Rudolph <lrudolph@panix.com>

Lee Rudolph wrote:
[above tentative sketch quoted in ful -- djr]

This is essentially correct.  If I get time in the next 
couple of days, I'll expand upon it.

--Chuck Giffen