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the (negative-3)-dimensional sphere

A 2-sphere ("2 dimensional sphere") is what you'd normally picture as a sphere, such as the surface of the earth. A 1-sphere is a circle. And a zero sphere is just a point. Higher dimensional spheres (hyperspheres) are also used a lot, both in math and physics (for instance, the large-scale structure of space in our universe most likely has the topology of a 3-sphere.) But today in Algebraic Topology he asked "do any of you know what a negative dimensional sphere looks like? say... a -3 dimensional sphere?" WTF?!

After a lot of blank stares, he asked the question again, followed by "c'mon, take a guess. be creative... anyone have any idea? Algebraic topology is so flexible, we can deal with negative dimensional objects very easily." Still a bunch of blank stares. One person then asked if it's the locus of points distance -1 from the origin. Clever, but obviously not right. "Nope, that's the empty set." He continued to ask us the same question... as if by asking us enough times, we'd suddenly go "aha! I can picture it now!" So he finally told us to go home and meditate on it. What a fun class! I'm still meditating, but I still don't have any ideas. Anyone know what he's talking about and want to give a subtle hint?

P.S. Got my colors done--haven't had a chance to take pictures yet.

Comments

( 12 comments — Leave a comment )
onhava
Apr. 11th, 2006 01:17 am (UTC)
I would suggest that you don't try to picture it, and keep in mind that algebraic topology is built on category theory, which is all about taking abstraction to a ridiculous degree.
onhava
Apr. 11th, 2006 02:23 am (UTC)
Also, here's the slightly less subtle hint: if I were to italicize a word or two in the preceding hint, things might be slightly clearer. ;-)
mauitian
Apr. 11th, 2006 01:47 am (UTC)
I think the answer is that he can't mean anything sensible. ;)

But, if I'm to guess at the particular craziness... the volume of a n-sphere (so, this would include the "surface area" for the familiar 2-sphere) is
V(n) = r^n (n+1) pi^((n+1)/2) / Gamma(n/2 + 3/2)
which is well defined for all real values of n except negative integers. I'd consider this a clue that the space in question is not compact. So, perhaps he's referring to a n dimensional space of constant negative curvature as a "-n-sphere"? But, nah, he probably means something weirder. But I don't think things like homotopy groups extend naturally to negative n.

Heh -- this is why mathematics can be bad for you. Ask this in your GR class and the prof should give you a dirty look.
mauitian
Apr. 11th, 2006 01:53 am (UTC)
Oops, that volume formula is well defined for n = -2, it only stops working at n = -3. For n=-2 it gives
V(-2) = - 1/(pi r^2)
So, err, you're on your own. Good luck with the weirdness.
spoonless
Apr. 11th, 2006 05:07 am (UTC)

But I don't think things like homotopy groups extend naturally to negative n.

Actually, negative homotopy groups is exactly the context in which he brought it up. He didn't say how they were defined either, but he assured us that there was a way of defining a &pi_-1 group, etc.

So I've mostly been trying to take that as a hint for how they might be defined. Unfortunately, the best I can come up with for making up a natural extension of what I'd want to define &pi_-n as... is to define it as the group of homotopy classes of continuous injective maps from a space backwards into an n-sphere (since the positive groups are defined as the forwards map). Seems to me this would make sense and give you some non-trivial information about how you can embed the space in other things. However, it doesn't help at all in defining a negative-dimensional sphere since it still uses positive n-spheres. So after coming up with that I threw my hands up.

The other idea I keep thinking of, is there must be some way of defining a Cartesian product of degree -n, some way of crossing the unit interval with itself a negative number of times. This would sort of line up with the hint cocacolaaddict gave (if indeed degree is the word he would have italicized), although even if that is the answer, I still have no idea how it would be defined. :) Ah well, if it doesn't come to me soon I'll just have to find out in class on Wednesday.
spoonless
Apr. 11th, 2006 05:38 am (UTC)

as the group of homotopy classes of continuous injective maps

oops, not sure why I wrote injective here, as the non-trivial part would be mapping larger spaces into smaller ones. I doubt this is the way it's defined anyway, though.
(Deleted comment)
spoonless
Apr. 11th, 2006 11:22 pm (UTC)

one way of defining it is to formally invert the suspension functor, and then define the negative spheres to be desuspensions of S^0 (which is two points, not one pont, btw).

Aha! This is most likely what he had in mind, as he had talked about suspension maps earlier that day. We haven't discussed them in a category theory context yet, but maybe he's getting to that. Funny thing is, I did briefly think of this... but I thought "nah, there's probably no easy way of defining it" especially since he only talked about going up in dimension and not the other direction (desuspension).

He hasn't used the word spectra yet either, but again... maybe it's coming soon. Perhaps he just wanted to taunt us with this early on before we get far enough that we're not as impressed.
mike_b
Apr. 11th, 2006 01:48 am (UTC)
there is a git.talk.math.topology , fwiw
wanton_adonis
Apr. 11th, 2006 02:13 am (UTC)
no clue, but if I had to design it, I'd flip everything inside out from its corresponding positive dimension.
onhava
Apr. 11th, 2006 02:22 am (UTC)
Speaking of turning spheres inside out: there's a fun result in topology about that. With a crazy video.
wanton_adonis
Apr. 11th, 2006 04:14 am (UTC)
sweet. I'd love to see a 3D viz of that!

can you flip a sphere inside out by putting a point force at the center that attracts everything inward, then momentum carries everything back out?
spinemasher
Apr. 11th, 2006 08:05 am (UTC)
Here would be my hint,

First of all, we have to recognize that the manner in which we first understand the question is absurd and cannot be answered. That is, if we understand the number of dimensions n to be the number of linearly independent coordinate axises then it's hopeless.

But then that's exactly the mistake. To conceive of a coordinate axis necessitates the existence of a coordinate system in the first place, which means we have a scale by which to measure things (at least locally). Now we know well that not all spaces fall into this definition. So throw it out!

So now what do we mean by "n" dimensions? Well clearly we are talking about some number perhaps an integer perhaps not. Recall that in dynamical theory we run into "fractional dimensions". So really all we want to do is count something and we want to generalize the way in which we count it. Typically, if we count something as being negative we mean a deficit. But more technically we mean the number which is the additive inverse of +1, so that under the binary operation +, we get the additive identity. Now focus on that for a minute together with what in the hell are we counting when we say "n dimensions".

Our next trick is to decide what properties are absolutely vital, that we insist we must keep. Do we keep the Abelian structure of the maps, do we keep isomorphisms (only? none?). In homotopy theory, one aspect of the theory is just how concerned we are with mapping transitivity. Should a triplet of maps with a single common space commute? That's the lift isn't it.

Can we go "backward"? Do the inverse maps exist? In essence we start to realize we are counting something like compositions but not quite because we are also concerned with whole sets of paths and any relevant (preservation of) group structures. Also we should decide what role the base point will play if any.

I hope this was helpful and not just more confusing.
( 12 comments — Leave a comment )

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