Defining a manifold without reference to the reals

The standard definition I’ve seen for a manifold is basically that it’s something that’s locally the same as Rn, without the metric structure normally associated with Rn. Aesthetically, this seems kind of ugly to me. The real line is a high-tech piece of mathematical machinery. We build up all that structure, then build the definition of a manifold out of it, then throw away most of the structure. It seems kind of like building an airplane by taking a tank, adding wings, and getting rid of the armor and the gun turret.

I’ve spent some time trying to figure out a definition that would better suit my delicate sensibilities, and have come up with the following sketch:
An n-dimensional manifold is a completely normal, second-countable, locally connected topological space that has Lebesgue covering dimension n, is a homogeneous space under its own homeomorphism group, and is a complete uniform space.

Does this work? I should reveal at this point that I’m a physicist, and no more than a pathetic dilettante at math. I have never had a formal course in topology. My check on my proposed definition consisted of buying a copy of Steen’s Counterexamples in Topology and flipping through it to try to find examples that would invalidate my proposed definition.

Since I’m not competent as a mathematician, what would probably be the best outcome of this question would be if someone could point me to a book or paper in which my idea is carried out by someone competent.

Clarification: I mean a topological manifold, not a smooth manifold.

Also, I should have mentioned in my original post that I had located some literature on the n=1 case in terms of characterizing the real line (which is not, of course, the same as characterizing a 1-dimensional manifold, but is a related idea):

P.M. Rice, “A topological characterization of the real numbers,” 1969

S.P. Franklin and G.V. Krishnarao, “On the Topological Characterization of the Real Line: An Addendum,” J. London Math Soc (2) 3 (1971) 392.

Brouwer, “On the topological characterization of the real line,” http://repository.cwi.nl/search/fullrecord.php?publnr=7215

Kleiber, “A topological characterization of the real numbers,” J. London Math Soc, (2) 7 (1973) 199

Answer

I’m going to risk an answer to this one. It’s a long answer, so I’ll
give a short summary first. One thing I’m not completely clear on is
whether you mean topological manifolds or smooth manifolds. If
you were a mathematician, I’d infer from your question the former, but
as you’re a physicist then I’m not confident of which.

There’s a big difference between the two cases and the answers are
very different. Here’s the short version:

  • Topological Manifolds I have considerable sympathy for your
    point of view, but have to say, “Get used to it.”. The point is
    that being a topological manifold is a property of a topological
    space and so is there whether you use it or not. We don’t study
    topological manifolds because it makes us look good, but because
    many of the “usual” spaces that one encounters happen to be
    topological manifolds. That they are topological manifolds means
    that we have a great toolbox to use to study them, but if we
    ignored that toolbox then the spaces would still be topological
    manifolds.

  • Smooth Manifolds Here I have less sympathy with your point of
    view simply because the real line is so integral to calculus. The
    real line might be an incredibly complicated gadget, but then it
    needs to be to support calculus. Of course, there are variants of
    calculus (holomorphic, p-adic) but if the real line looks
    complicated, then I would be amazed to hear that the complex plane
    or the p-adics looked any simpler. Nonetheless, because being a
    smooth manifold is about structure, it is actually more feasible
    to entertain different definitions.

Okay, that was the short version. Now for the long version. First, I
need to say something about definitions.

Mathematical Definitions

You say that you are a physicist, so it’s possible that you haven’t
been let in on the secret about mathematical definitions. If you
have, skip this bit. If not, I’ll tell you. (But, hush! It’s a
secret. Don’t tell anyone else.)

I’ll illustrate the point I wish to make with an example that I hope
is familiar to you. What is the definition of a continuous map
between metric spaces? I teach this, and I teach three
definitions:

  1. A function f:MN is continuous if whenever (xn)x in M then (f(xn))f(x) in N. (Note: I chose metric
    spaces here, so sequences are sufficient.)

  2. A function f:MN is continuous if for every xM
    and ϵ>0 then there is a δ>0 such that whenever
    dM(x,y)<δ, dN(f(x),f(y))<ϵ.

  3. A function f:MN is continuous if whenever UN is open then f1(U) is open in M.

These definitions are equivalent: they all agree which functions
are continuous and which are not. So any statement made using one
definition can be reformulated using another. But they have different
uses, since they emphasise different aspects of what it means to be
continuous. If you’re interested in metric spaces because you use
approximations then the first definition captures the idea of what you
want to use: If I have an approximation of something, then after I
hit it with a continuous function, it is still an approximation (of
the image of the “something”). The second definition is actually the
most practical when testing an explicit function for
continuity: it’s amenable to finding estimates and the like. Third is
the most theoretically powerful: it’s the first one we reach for
when trying to prove theorems about continuous functions.

So just because a definition seems to be the “established” definition
doesn’t mean that that is the right way to think about it.
Definitions are malleable, and we often use a different definition to
the one that we truly believe is “right” simply because it is easier.

Topological Manifolds

Let me start with topological manifolds. I said at the beginning that
the key here is that being a topological manifold is a property.
That is, if I have a topological space then it either is or isn’t a
topological manifold. If it is, then I can use that fact when I study
it; if it isn’t, then I can’t. If it is a topological manifold then
I don’t have to use that fact, but I’m likely to be making life
difficult for myself if I don’t. The key thing about a property is
that if I ignore it, it is still there.

The people who study topological manifolds do so because many spaces
of interest happen to be topological manifolds. If you change the
definition, then those spaces will go on being locally Euclidean, and
the people studying them will continue to use the fact that they are
locally Euclidean, and all that will have changed is the language that
they use. This is why I don’t have much sympathy for your desire to
change the definition.

Although most of the Euclidean structure doesn’t have much
topological influence on a topological manifold, it does provide a
lot of useful tools in the analysis. For better or worse, Euclidean
spaces are things that we simply know a lot about. So saying that a
space is locally Euclidean means that we can use all our intuition and
skills from the theory of Euclidean spaces to the study of the space.
That’s worth a lot, and you’d need to be very convincing to persuade
people to give that up.

Now when thinking how to define a topological manifold, one encounters
the question I alluded to in the above on definitions. Definitions
come in all shapes and sizes. It’s not clear from your question as to
which definition you would like best. On the one hand, your dislike
of the real line makes me think that you want the “pure” definition:
the one that captures the soul of a topological manifold. I’ll
readily agree that the current definition is not that, it’s more of
the “body” type where it’s easy to see how to use it. But the
proto-definition that you give isn’t that either: it’s a mish-mash of
topological concepts, each of which excludes a range of spaces, with
the hope that in the end all you have left are the topological
manifolds. I don’t like that sort of definition, it’s more of the
ϵδ type: has its place, but is neither the “soul” nor
the “body”.

However, what it feels most like is that you are playing that
children’s game where you have to explain what is an aardvark without
using the words “aardvark”, “anteater”, “dictionary”, or “pink
panther”.

An alternative is to come up with a definition that is actually
different in that it doesn’t completely agree with the current
definition. In that case, your work is harder. You have to show why
the new definition is better than the old one. The most convincing
arguments would be either that your definition allows you to do more,
or that it allows you to consider more spaces. But these are unlikely
to both hold. If you allow more spaces, you probably lose out on
abilities; if you find new tools, then you probably can’t apply them
to all the current topological manifolds. If you really want to do
this then your best bet is not to mention topological manifolds at
all, but to invent a wholly new concept, say “Topological foldimen”
and simply say, “Topological manifolds that are X are foldimen, and
foldimen that are Y are manifolds.”. Then hope that there are
plenty of interesting foldimen out there.

Smooth Manifolds

Smooth manifolds, on the other hand, are much more malleable. This is
because being a smooth manifold is something a little bit extra.
The standard definition of a smooth manifold starts with a
topological manifold and then adds a little extra on top. Now,
forgetting that extra does mean something. If you forget it, it goes
away, and you can’t be sure what it was.

As an illustration, if I have two topological manifolds, X and Y,
then the question “Is X homemorphic to Y?” has the same answer if
I remember that they are manifolds or not. But if I say that they are
smooth manifolds, then the question “Is X diffeomorphic to Y?”
depends completely on their being smooth manifolds.

This actually gives us some room to manoeuvre. Because we need to
construct the extra structure, we can consider different
constructions. However, this is where your dislike of the reals
counts against us. We cannot construct something from nothing: we
need to start with something. There are many possible answers as to
what that “something” is, but they all boil down to identifying
certain spaces as “known” meaning that we decide what the structure
for those spaces should be. Since they are “known”, and everything
else will be defined relative to them, they should be spaces that we
really do know about. It’s hard to get spaces that are more
well-known than the Euclidean spaces. Certainly when calculus is
concerned. All of the examples of this that I’ve seen have used
Euclidean spaces, or “nice” subsets thereof.

So these are our “known” spaces, which we will use to define what a
“smooth structure” means for “unknown” spaces. This is where we have
some flexibility in the definition, and here is where we can get rid
of that annoying “local stuff”. Maybe, just maybe, we don’t need to
have actual charts and can get away with something weaker.

Actually, we can. No “maybe” about it. But the problem is that by weakening the
definition, we end up with more things than we might like to admit to
the hallowed halls of manifolddom. Nonetheless, there is some merit
in pursuing this line as it separates out the construction aspect (of what “smooth” means) from the property aspect (of what a “manifold” is).

As I said, there are many approaches at this point. I’m going to
outline one just so that you have one in mind. There are others, and
this is not the place to evaluate them. If you don’t like this one,
the rest still carries through. I just want to be sure that you have
one picture in your mind.

Here it is: it rests on the slogan that manifolds are all about smooth
curves. If we both look at a manifold, we should agree on which
curves in it are smooth and which are not. So one way to specify a
“smooth space” is to give a list of all its smooth curves. One
probably wants some conditions, but this can all be made precise.
Thus a “smooth space” is a space in which we all agree on smooth
curves.

As I said, there are other approaches, but whatever they are they
still give us a list of the smooth curves. This will be important in
a minute.

Now it turns out that this admits far more than just manifolds. In
fact, too much. There are really weird spaces in our list now, and
we’d like to get rid of them.

Remember the curves? Good. We’re going to use them. Using the
curves, one can define the tangent space of a “smooth space”. Just
the same as for a smooth manifold: derivatives of curves. Unlike
smooth manifolds, this needn’t be locally trivial, nor even the fibres
vector spaces.
There’s also a notion of a topological tangent bundle, which is
related to neighbourhoods of the diagonal.

Here’s the definition:

A smooth manifold is a smooth space whose smooth and topological
tangent bundles agree.

Now, I’m not 100% sure that this is exactly the same as “smooth
manifold”, but certainly all smooth manifolds fit this description,
and it excludes smooth manifolds-with-boundary, but there might be the
odd pathological case that this doesn’t exclude. Nonetheless, it is a
very powerful description: it implies that the tangent spaces are
actual vector spaces – we didn’t assume that, if you remember.

Conclusion

If you want to mess with definitions, go ahead: it’s fun. But be
careful that you know what type of definition you are aiming for.
There are reasons that the current definitions are what they are and
the reasons tend to be pragmatic: over the years, we’ve found certain
definitions useful and others not. There is a lot of dead wood, and
there are always new insights that shed new light on old concepts, but
if you want to replace an old definition, then remember that it’s been
there for a while, working hard, and will strongly resist all attempts
to “retire” it from service!

Attribution
Source : Link , Question Author : Community , Answer Author : Andrew Stacey

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