From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Klein bottle question
Date: 18 Oct 1998 08:10:25 GMT
Kate Orman wrote:
>Ah... so the folks living on a Klein bottle wouldn't be three-dimensional
>like us, they'd be two-dimensional, like the Flatlanders dwelling on the
>Moebius strip - is that right?
May I suggest that the ideal of "dimensionality" bandied about here is
a bit murky?
As has been pointed out in other posts, most of the spaces being discussed
are (intrinsically) two dimensional. Roughly speaking this means that
from any point in the space one describes the location of nearby points
by giving two "coordinates" or "displacements"; note that this only
works locally, and that it requires the establishment of a coordinate
convention. Let me give some examples of two-dimensional spaces.
The plane is two-dimensional. At each point in the plane you can draw
two vectors which can be used for directions near that point. When the
plane is the blackboard, we might have a vector pointing right and another
pointing up. Observe that in fact the coordinates established at one point
suffice to determine positions everywhere in the space: every point is
unambiguously located by telling how far right and up it is from the
origin (or any other pre-assigned Mecca).
A torus -- the doughnut (or coffee-cup :-) ) -- is a two-dimensional
manifold. At each point we can draw a pair of vectors tangent to the
surface -- say, a red one pointing one way parallel to the "rim" of the
torus, and a blue one pointing along its "waist". Now, I can pick one
location to be the "origin" again, and tell how to reach any other point
by travelling a certain distance "around" (along the red) and a certain
distance "over" (along the blue). This time, however, a point does not
have a unique set of coordinate; the same point can be reached by
going 1/2 unit "blue" or by going -1/2 unit "blue" (where the distance
around is taken as the unit of measure.) We say the torus is not
homeomorphic to the plane (although it is locally homeomorphic to it).
The earth is two-dimensional. We give coordinates at most points by
specifying displacements "North" and "East", say. Now we have to surrender
another feature: it is impossible to establish directions uniformly
across the earth (there is neither "north" nor "east" at the poles). This
is a fundamental problem: you cannot set up two sets of nonzero-vectors
(red and blue, say), one set at each point. Indeed, one cannot even draw
a _single_ nonzero vector tangent to the sphere at each point, varying
in a continuous way, let alone trying to draw two which never point in
parallel directions. This is the "Hairy Ball Theorem" (You can't comb
the hair on a sphere). The property which the torus had but the earth
lacks is parallelizability.
Note however, that we could set up coordinates
across the earth something like this: take a snapshot of the earth from far
above the north pole, and mark off blue vectors pointing right (towards the
Greenwich meridian, say, more or less) and red vectors pointing up
(more or less towards the Indian Ocean). Think of how these vectors point
at locations a little north of the equator: in Ouagadougou, the blue points
south and the red points east; in Calcutta, blue points west and red points
south; in Honolulu, blue is roughly northeast and red is roughly northwest.
You could extend these definitions of red and blue directions to neighboring
points just south of the equator. You could also cover the whole
southern hemisphere with a similar set of vectors. This would make for
sort of a confusing situation near the equator; ships crossing that line
would have to take note of the fact that they are switching from
north-hemisphere convention to south-hemisphere convention. But it wouldn't
be all that bad: strip maps of the equator would show _two_ sets of
arrows, one red and one blue in each set. The key observation is that one
could convert one set at a point to the other set simply by _rotating_.
(For example the conventions might be in synch at the dateline and prime
meridian, but 180-degrees out of phase in Quito.)
A Moebius strip is a two dimensional manifold, too. This time, too, we
draw a (non-parallel) pair of vectors at each point. We _can_ comb the
hair in the strip -- at each point, draw a blue vector along the "center
circle" of the strip. We can't draw a second direction uniformly, as in
the torus, but certainly at each point we may pick one of the two
directions "crosswise" as the red vector. Indeed, with a pair of "crosswise"
cuts we can split that pesky Moebius strip into two rectangles, which
we can call the "northern" and "southern" ones, each with its own uniform
conventions of blue and red directions. Again, maps of the "equator"
(that is, the two regions where we made the cuts) would have to have
two pairs of colored arrows at each point. Without any real trouble, they
could be made to agree completely along one of the two cuts. But, and
here's the kicker, along the other part of the equator we would find
the sets of arrows to be "flipped": if the two candidate blue vectors are
rotated to point in the same direction, the red vectors would still be
180 degrees out of kilter! The property we have lost from the other
models is orientability.
(Actually we lost another property: edgelessness. If it is possible to
move at a uniform speed across the Moebius strip, then in a finite amount of
time we encounter a "boundary of space"; if not, then the Moebius band,
like the plane, appears to "go on forever". The torus and sphere, as well
as the Klein bottle, are different: they are compact, meaning a traveler
on them experiences no boundary, but if travelling long enough will eventually
come close to a point previously travelled.)
The Klein bottle is in fact nothing more than two Moebius strips joined
together. Each of them has a boundary which is a single circle, so two
identical Moebius strips can be stitched together as if zipping up a zipper
along two identical lengths. Now, I'm a professional -- don't try this at
home. There is a wee bit of a problem as you get near the end of this
tailoring job. It seems the things the zipper halves are attached to
are rather cumbersome, and seem to get into each other's way. This
is an embeddability problem, which I'll come to next.
Likewise one more space, while I'm at it: simply take a disk, and sew
each point in its boundary to the point directly opposite it. This is a
two-dimensional manifold, compact, and non-orientable. It's the "projective
plane" or "cross-cap".
This rounds out my display of two-dimensional surfaces, but there are more:
two tori can be stitched together if you remove a little circular patch
from each; the result is a "two-holed torus". Similar attachments allow
the construction of grander spaces; in fact this is how all the compact
surfaces may be constructed.
Now let me discuss this issue of "living on" a surface. I have been careful to
refer to points _in_ these spaces, rather than _on_ them. That's because the
fundamental properties of the spaces have to do with the points of the space
and their neighbors in the space; what happens around them ("in the air" if
you like) is not part of their topology. That's a slightly different question,
called the embedding of the space into another. You may have seen a trefoil:
a kind of loop which winds around itself three times in a knotted way,
returning to its starting place. Intrinsically, that's just a circle. We
say the trefoil and the circle are homeomorphic (the points within the spaces
can be paired off appropriately) but differently embedded into space
(the points of the Euclidean spaces around the circle and trefoil cannot be
paired off in a continuous way). We have described the Klein bottle and
cross-cap, but there is no way to embed them into 3-dimensional space.
This "instrinsic" view of the space is often described in terms of its
inhabitants. We live on a sphere but it looks like the plane near
where we stand. The same is true for inhabitants on any other 2-dimensional
manifold, if those inhabitants are constrained to looking only along
the "red" and "blue" directions and the plane they span; this lets
them look only at nearby points in their space. (A space-dweller living
in a toroidal space station -- a popular design, I take it -- is assumed to
look only left/right or forward/back; looking straight up at the "ceiling"
isn't allowed). This is the perspective taken in the classic "Flatland".
Notice the Flatlander living on the Moebius strip is without recourse to
reconcile the two coordinate systems on the halves of the strip: turned
to face the mutually-agreed upon direction of "blue", s/he will have
the northerner's "red" to the left and the southerner's "red" to the right.
On the other hand,we are in reality 3-dimensional beings. We _could_ look up
inside a toroidal space station. We _do_ look down at our spherical earth
and see it's not a plane. Forced to walk around a Moebius strip like
Escher's ants, we could reconcile the coordinate systems there by
(continuing to face "blue" and) standing on our heads when we need to
translate from one system to the other. (Actually we would walk on it and
find ourselves "on its other side", a concept foreign to the Flatlanders.)
And on the Klein bottle ...
...well, what? Here the concept of "walking on", as opposed to "walking in"
has no meaning. We cannot really walk on something which is not embedded
in R^3. To understand the problem here, imagine the Flatlanders' pet
worm. It is accustomed to walking around a circle in Flatland. Yet if we
embed that circle into 3-space, the worm no longer has a notion of which
way is "out" on the circle, and might spiral around the cross-section
as it crawls around the circle, coming back to its starting point on the
"inside" of the circle. What was a simple task on the circle in the
plane is now harder to carry out the same way on a circle in 3-space.
What is needed is a framing, that is, an additional vector at each point
of the circle, pointing "out", so the worm knows how to orient its
body relative to the circle at each point. So too it is impossible for
us to walk "on" a surface embedded in R^4 (say) unless the surface is
also equipped with a third direction at each point. Indeed, should you be
given a chance to embed the earth in to R^4 and then wander around the earth,
you could easily come home staring into your basement instead of your front
door, if you don't take the precaution of embedding an "out" vector
along with each point. Your poor 2-dimensional retina would undoubtedly
be overwhelmed by the opics in R^4 but I imagine you would only see a
circle's worth of the spherical earth at each moment, until you're suddenly
confronted by your laundry room. (Intrigued? There is a film worth watching:
"Turning a sphere inside out"!)
If the business of embedding is too dis-orient-ing (sorry) we can mimic the
Flatlander description of topology by discussing three-dimensional manifolds.
These are just spaces in which there are three independent directions,
say "red', "blue", and "green" at each point. Your living room is one
such example. So is the inside of the space shuttle: note that we are not
requiring any preferred direction to be called "up". A thickened Moebius
strip is an example of a 3-dimensional manifold: simply take "red" and
"blue" as before, and pick "green" at each point to be a vector
pointing straight away from the Moebius strip into space. As with "red",
this vector cannot be picked in a continuously-varying way across the whole
of the space, but that's OK.
Now, topologists recognize that the thickened Moebius strip is nothing but
a solid torus. Launch one into space and ask people to take a walk, and
the worst that would happen is that they would return home upside down
relative to when they left. These kinds of things happen all the time in
space, and Miss Manners has learned to be patient.
Far more interesting is the direct product M x I (M=Moebius strip,
I=unit interval). For the cognoscenti: this is almost the same as
above, but the identification on I^3 is not (0,s,t)~(1,-s,-t), as in
the previous paragraph, but rather (0,s,t)~(1,-s,t). This construction
uses a reflection, not a rotation. As a result, this space resembles a
long curved indoor racetrack which loops back on itself in a funny way: when
you come back to the starting line, the left lane ends where the right one
began and so on. Did you start at the left lane and shake you trainer's
right hand just before you started down the track? Then upon your return
your trainer will appear to be at the right-most lane, extending her
_left_ hand for a congratulatory handshake! (Obviously "left" and "right"
have no more meaning in this world than "up" and "down" do in deep space,
and just as obviously, I suppose, this manifold does not embed into
Euclidean 3-space). By the way, if your left hand hurts, just run around the
track again: former positions will be regained and you can shake with your
right.
There are some variants of this construction which are also fun.
Projective 3-space has been mentioned. If it's big enough, it feels
like Euclidean 3-space, but when you use a good telescope, you see
the back of your head, and it's upside-down. Not only that, when you look
straight up, you see your shoes, turned to look as if they belonged to
someone talking to you, face to face. For more fun, consider the
Poincare sphere; this lets you see anything around you from the "outisde"
perspective by looking in any of 120 directions. (This space is X=S^3/G
where |G|=120 is perfect, so that X is a homology sphere but not
even simply connected.)
A few respondents mentioned geometry, which I haven't yet since the issues of
orientability (etc.) are at the more fundamental topological level.
But in fact one can get some interesting geometry here too. One can
endow a manifold with a construct known as a Riemannian metric; this can
then be used form computing lengths, areas, and volumes. I mention this
only to say that the metric need _not_ be inherited from R^3 or any other
Euclidean space containing the manifold. For example, there's no difficulty
in assuming that the Moebius strip is infinitely long "crosswise". Also
curvature is not forced by the topology except at a really basic global level.
For example, the torus, which we usually think of as a curved subspace of
R^3, may be embedded into R^4 as a "flat" torus (S^1 x S^1). Our
understanding of space time is that it is indeed unevenly curved; but
we have no evidence that it is not topologically trivial.
Topics of this type have arisen before. For differential topology
(that's manifolds, knots, etc.) see
http://www.math.niu.edu/~rusin/known-math/index/57-XX.html
For differential geometry
http://www.math.niu.edu/~rusin/known-math/index/53-XX.html
dave
==============================================================================
From: "Dr. Michael Albert"
Newsgroups: sci.math
Subject: Re: klein bottle question
Date: Tue, 20 Oct 1998 22:41:22 -0400
Dear Friends:
A kind person has pointed out to me that if you take a solid ball and
identify antipodal points on the surface, the result is indeed
orientable. My apologies for the confusion. I had just wanted
to point out that non-orientable spaces came in sizes bigger than
two dimensions, but the example I gave was of course completely wrong.
As this person has not posted, I assume that it was due to a wish for
privacy. However, there is little I can do to improve what I was
sent, so I am taking the liberty of quoting it at length:
> In one of your postings to sci.math (10/17), you described RP^3
> as being an example of a non-orientable compact 3-manifold.
> However, it is orientable, since H^3(RP^3, Z) = Z. The difference
> between this and RP^2 is that the antipodal map on S^3 is orientation
> preserving, whereas the antipodal map on S^2 is orientation reversing.
>
> (In fact, RP^3 is homeomorphic to the Lie group SO(3), and one can
> translate a volume element at the identity around the whole group to
> obtain a nowhere-vanishing volume form.) RP^2 x S^1, or Klein bottle x
> S^1, would have worked instead.