From: "Michael Weiss"
Subject: Re: Diffeomorphisms 'n stuff.
Date: 8 Jun 2000 22:47:05 GMT
Newsgroups: sci.physics.research
Summary: [missing]
Me:
>So "straight line" is a projective property.
Oz:
| Er, yes. Mind you isn't that because 'flat' and 'projection' are
| things that you make from straight lines? Wouldn't it be better to
| class them in the bunch of stuff that is made from straight lines in
| some way?
Sure, if you like. Though I wouldn't worry about which is prior,
projections or straight lines.
If this were a textbook-worthy discussion, I'd have to be far more
careful with my definitions. In formal treatments, circular
definitions are a no-no. Defining "straight line" in terms of
projective transformations, and "projective transformations" in terms
of "straight line", puts you in danger of a visit from the Cyclic
Ordering Police. Likewise for "continuity", "neighborhood", and
"topological property".
But there's more to math than Definition-Theorem-Proof. Most folks
don't want to just follow the math, they want to understand it. And
for that, it helps to *wander around* a bit, like exploring a new
museum. The builders have to construct the edifice brick-by-brick
from the bottom up. But we can ramble from floor to floor, glancing
first at the condors, then at the pterodactlys, then back to the
condors, before climbing up two floors to look at the plane in the
wind tunnel.
That's what I'm trying for here: a leisurely stroll, seeing
the connections among the concepts, without worrying about
arranging them in a strictly logically sequence.
| Generally I tend to handle this by picking an object which patently
| exhibits the required characteristics and patently excludes the non-
| required/disallowed characteristics. I then hold this as a 'class
| example'.
A good strategy sometimes --- but with some drawbacks. Let's say
you decided to buckle down and learn the precise mathematical
definition of a topological space. No doubt you are expecting
something fearsomely complicated, with clauses and subclauses galore.
You are at first pleasantly surprised to see how short and sweet the
formal definition is:
Definition: a topological space consists of a set X, whose elements
are called points, together with a collection of subsets of X, called
open sets, satisfying three conditions:
Top1: The empty set and the set X are both open.
Top2: The intersection of any two open sets is open.
Top3: The union of any family of open sets is open.
Then you do a double-take. "Huh! What does it *mean*? Can you give
me an example?"
So we trot out a standard example or three: the real line, the
Euclidean plane, the sphere. Now *all* of these examples come
*dripping* with extra mathematical properties, above and beyond being
topological spaces. On the other hand, these three examples have a
lot to tell you about the *meaning* of topology --- about what it has
to do with continuity, and why mathematicians dreamed up the
definition in the first place. If we started out with messy
pathological examples, you'd have a much harder time learning
your way around.
| I might consider a homeomorphism as two jaggedy loops side by side
| with a reversible arrow between.
Not a bad picture.
| I might consider a homotopic map as a sort of tube from one jaggedy
| loop to another with the intervening (ie continuous bits) making the
| tube and a single-ended arrow in it.
Even better. But put these two visual aids side by side, and you're
ripe for confusion. I'll say more about that in a moment.
You can also imagine a *movie*, with one jaggedy loop morphing into
the other one. The movie can't have "jump cuts". (It's a square!
It's a circle!) But it can have sudden changes in tempo,
like you often see in commercials these days. (It's qckly mrphng ---
it's v-e-r-y s-l-o-w-l-y m-o-r-p-h-i-n-g.)
| Ok so a homeomorphism is a continuous isomorpism
A homeomorphism is an isomorphism in the category of topological
spaces. The formal definitions of 'topological space' and
'homeomorphism' were first given decades before category theory was
invented. That's why we have the special term 'homeomorphism'.
Indeed, many notions in category theory embrace dozens of special
cases of older vintage. Category theory was one of those Grand
Unifications that math goes through every now and then.
| but a homotopy is not necessarily an isomorphism?
Here's where your two pictures, each good by itself, taken together
are tripping you up. I see that Toby has already chastised you on
this point: homeomorphisms go between topological spaces, homotopies
go between maps.
(To make things more confusing, there is a notion of two spaces being
in the same *homotopy class*. People sometimes say the spaces are
homotopic when this happens, though the official term is 'homotopy
equivalent'.)
In categorical terms, an isomorphism is a reversible morphism, and a
morphism goes from one object to another. A homotopy goes from one
*morphism* to another (in the category Top), so it must be a different
kind of beast. Indeed, it's a 2-morphism in a 2-category. John Baez
and jim dolan (and others) have been hard at work on a wonderful new
exhibit, "All About N-Categories"--- you might want to stroll through
that sometime.
But let's stay right here in the Great Hall of Topology for a bit. We
put on our earphone-guide. Here in an exhibit case titled
"Homeomorphisms", we see two jagged loops, labelled X and Y, with an
arrow between them. Let's bend down and read the plaque. "These
loops represent topological spaces. Do not think of these loops as
being *in* 3-space. An inhabitant of loop X cannot get *off* the loop
and move through the surrounding space --- there *is* no surrounding
space. However, he can use the homeomorphism (represented by the
arrow) to *teleport* himself instantly to space Y. Because
homeomorphisms are continuous, he may get stretched, but he won't be
torn."
We walk over to the next case, titled "Homotopies". Here we see
jagged loops labelled f and g, and a tube (labelled h) between them.
The f-loop is bright red, the g-loop is blue, and the tube is a
silvery grey.
The plaque says: "The jagged loops f and g represent mappings from the
circle C into 3-space. Really the loops are the *images* of the
mappings." The earphone murmers, "They don't mean images in the
colloquial sense. They mean the formal math definition: if f maps A
into B, then the image of f is the set of all points in B of the form
f(a), for some a in A. That is, it's the set of all *targets* of the
f-arrows."
The plaque continues, "The mappings both map a circle space into
3-space. Press the red button to see f mapping, and the blue button
to see the g mapping." We press the red button. The tube and the
g-loop disappear. Somewhat above the jagged f-loop we see a bright
glowing white circle, like a halo. Ghostly red arrows, each labelled
with a pale f, go from points on the halo to points on the jagged
f-loop. We press the blue button, and a similar 3d image appears,
with blue arrows pointing to the g-loop. We press both arrows at
once, and we see both loops with both sets of arrows, but no tube.
"Now press the white button to see the homotopy h." We do so. The
white halo, the red jagged loop, and the red arrows are now visible.
The tube is also faintly visible. The red arrows move down the tube,
targetting a bright jagged cross-section of the tube. The
cross-section and the arrows gradually change from red to blue as they
move along, leaving a silvery grey tube in their wake. The exhibit
now looks as it does when we first laid eyes on it.
Finally we wander over to an exhibit case in the corner. It is
titled, "Cobordant Submanifolds". It looks much like the homotopy
case, except the loops are smooth, and are labelled X and Y. We read
the plaque. "X and Y are submanifolds of a containing differential
manifold, here 3-space. The grey tube is a cobordism between them.
Press the button to see another cobordism." We press the button. The
red loop remains, but the blue loop is replace with two smaller blue
loops. Also, the silver-grey tube is replaced with something that
looks like a smooth pair of pants. The red loop is the waist, the
blue loops are the cuffs.
"Hmm," you mutter, "so is the red loop homotopic to the blue loops or
not?" As if on cue, the earphone chimes in: "Compare this exhibit to
the exhibit of homotopies on your left. The red loop and the pair of
blue loops are not mappings. We could *construct* a mapping from a
circle to 3-space, whose image would be the red loop. However, we
*cannot* construct a mapping from the circle to 3-space, whose image
would be the pair of blue loops."
The plaque continues, "Pontryagin used the concept of framed cobordism
to determine the smooth homotopy classes of maps from a compact
n-dimensional differential manifold into a p-dimensional sphere.
Press the white button to see a video." We press the white button.
Images appear in profusion: spheres, surfaces with little arrows all
over them, morphings and special effects galore. The earphone murmurs
soothingly, soporifically. We wander off towards the exhibit titled
"Fundamental Groups"...