Date: Thu, 12 Nov 1998 14:46:11 -0800 (PST)
From: Alan Weinstein 
Subject: improper action

 John,

 Could you please forward this message to all the gfest attendees?
 Thanks again for organizing the affair.

 Alan
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Dear Groupoidfesters,

It was great to see all of you last weekend.   Here is a postscript to
my talk:

Marc Rieffel showed me an example of a free action of R on a manifold
which has a Hausdorff orbit space, but which is not proper.  Here is
my version of it.

The Mobius band admits a well known action of the circle, with one
orbit half as long as all the others.  To make this action free, we
remove one point from each orbit by cutting a slit from the edge of
the strip, transverse to the orbits, until we reach the exceptional
orbit which runs down the middle.  The resulting vector field is not
complete, but we can make it complete by multiplying by a function
which goes to zero sufficiently fast as the slit is approached.  For
this R action, the quotient space is the same half-line as before,
certainly Hausdorff.  But since the orbits slow down more and more as
one approaches the slit, any open set which crosses the exceptional
orbit must return to itself after arbitrarily long times, which makes
the action improper.  

What we must have, then, is a closed equivalence relation on the slit
Mobius band M (still topologically a Mobius band) which is the image
in M x M of a 3-dimensional manifold Q (the action groupoid) under a
non-proper embedding.  How can the image of a nonproper embedding be
closed?  The simplest example I know is the figure eight in the plane,
realized as an embedding of an open interval.  (The two "ends" of the
interval are at the point where the "eight" crosses itself.)  I
wouldn't be surprised if this simple example were to be embedded
somehow in the more complicated one above.

Best regards,

Alan