🚀 go-pugleaf

I would like to recommend Jeff Week's "The Shape of Space" book as an 
excellent elementary introduction to manifolds.  And I'm not kidding
when I say it is an excellent elementary introduction to manifolds --
you could use this book to explain fiber bundles to high schoolers.
This book does not suffer from the calculus-test disease that John 
Baez refers to (the ambitious urge to throw in everything having to
do with manifolds), but in all fairness, you really do need to see 
lots of manifolds and their applications to get a feel for them.  Just
giving the standard open-neighborhood-looks-like-R^n doesn't do a lot
for most people.

I see that we have sci.physics on the Newsgroups: line, so I would
like to take this opportunity to relate a bad pun.  A couple of years
ago some astronomers did a statisitical star count survey.  This survey
was a predecesor of those which found the Great Wall, the Great Void, and
other fascinating large scale structures, but it differed in that it took
a much narrower ray and followed it out much farther.  Well, when the
astronomers plotted the number of stars they found as a function of distance
they were surprised to find that the distribution was almost periodic.  
That is, if you take into account the fact that the ray measures out a
bigger area the farther you go out, the star count graph would almost
repeat itself.  Well, when the mathematicians found out about this, they
were very excited, as this would be exactly what should happen if our
universe was a non-trivial three dimensional manifold -- that is, if our
universe had loops.  In particular, one of the simpler and thus perhaps
more likely manifolds for which this could happen was discovered by Jeff
Weeks (the author of the above recommended book).  Thus would the ancient
prophecy of the universe being made in a Weeks be fulfilled.

-Thomas C