What is a Product of Spheres?

[Jooolz]

Hi all,

I would really appreciate if someone can explain to me what is meant by a product of spheres.
What would for example S¹ x S⁰ look like? The first being a circle and the second being a pair of boundary points... So what kind of "object" is their product?

And how about S¹ x S¹ or S¹ x S²

I hope someone can clear this up for me.

Kind regards,

 

[I like Serena]

Hi all,

I would really appreciate if someone can explain to me what is meant by a product of spheres.
What would for example S¹ x S⁰ look like? The first being a circle and the second being a pair of boundary points... So what kind of "object" is their product?

And how about S¹ x S¹ or S¹ x S²

I hope someone can clear this up for me.

Kind regards,

Welcome to PF, Jooolz!

S¹ x S⁰ is a circle that is combined with each of the boundary points.
Consider a circle in the X-Y-plane.
And let the boundary points be at z=+1 and z=-1.
What you get is 2 circles.

S¹ x S¹ is a circle, and at each point of the circle we have another circle that extends in a dimension.
That's... a donut!

S¹ x S² is a bit more difficult, since we can't visualize it in 3D anymore.
It's a circle with at each point a sphere in a different dimension.
That's a kind of hyperdonut.

 

[Jooolz]

thank you very much helping! :-)

I don't think I really got it... So S¹ x S⁰ = S¹ x {-1, 1} ?
with -1 and 1 on the x-axis? then draw two circles.. with midpoints (-1, 0) and (1, 0)? Or does it not make sense to talk about -1 and 1 on x-axis cause there actually isn't one...cause -1 and 1 are just some loose points that could've also be named A and B??
AAARRRRRGGG I am making myself more confused now... :-S

How about S¹ x ℝ?



kind regards,

 

[I like Serena]

thank you very much helping! :-)

I don't think I really got it... So S¹ x S⁰ = S¹ x {-1, 1} ?

Yes.
This is isomorphic to 2 circles parallel to the X-Y-plane with their midpoints at (0,0,-1) and (0,0,+1).
Or if you want you could pick midpoints ((0,0), -1) and ((0,0), +1) which matches the cartesian product.

How about S¹ x ℝ?

This is (isomorphic to) a cylinder that is infinitely long.

 

[Jooolz]

Of course... it is a cylinder :shy:

Can you tell me if I'm thinking in the right direction with this:

If we define an equivalence relation on S¹ X ℝ by

(x, t) ~ (-x, -t)

Do they mean, that if we choose a point x on the cylinder (which is a point on a circle) the t would be automaticaly the midpoint of that circle? And then that point must be identified with (-x, -t) ? I tried to pictured it... but the -t part doesn't make sense to me...

first this: is it true that if we have a circle, and the equivalence relation x ~ -x that we would end up with half a circle?

I so much appreciate that you take time to help me.. Thank you!
I like Topology, but god it is not easy...

kind regards,

 

[I like Serena]

Of course... it is a cylinder :shy:

Can you tell me if I'm thinking in the right direction with this:

If we define an equivalence relation on S¹ X ℝ by

(x, t) ~ (-x, -t)

Do they mean, that if we choose a point x on the cylinder (which is a point on a circle) the t would be automaticaly the midpoint of that circle? And then that point must be identified with (-x, -t) ? I tried to pictured it... but the -t part doesn't make sense to me...

"t" would not be a midpoint of that circle, (0, t) would be the midpoint of a circle.

x would be a point on the circle, which would actually be (x,y).
t would be the z coordinate.

Your equivalence relation would identify each point of the cylinder with the point on the opposite side of the origin.
I'm getting a headache trying to visualize that.

first this: is it true that if we have a circle, and the equivalence relation x ~ -x that we would end up with half a circle?

Hmm, I dunno.
Never did something like that before.
Will you let me know if you find out?

 

[micromass]

first this: is it true that if we have a circle, and the equivalence relation x ~ -x that we would end up with half a circle?

Nope, you will end up with a very weird space: a projective space. http://en.wikipedia.org/wiki/Projective_space

This space is extremely important in geometry and algebraic topology.

 

[Jooolz]

"t" would not be a midpoint of that circle, (0, t) would be the midpoint of a circle.

x would be a point on the circle, which would actually be (x,y).
t would be the z coordinate.

I was thinking of it this way: having the infinite cylinder standig so that the height is the z-axis... and then a point of a circle would be (x, y, z) where z is the midpoint...Cause the height of points on the same circle would be the same... Wrong?

Your equivalence relation would identify each point of the cylinder with the point on the opposite side of the origin.
I'm getting a headache trying to visualize that.

So Am I... :-s

 

[Jooolz]

Nope, you will end up with a very weird space: a projective space. http://en.wikipedia.org/wiki/Projective_space

Now you've made me cry...

In my book they are saying that the one-point-compactification of the quotiënt space (S¹ X ℝ)/~ Is in fact homeomorphic to the real projective plane...

So of course I want to know why that is... pfffff... first I wanted to know what I am actually dealing with.. And now I don't like it... cause I really want to understand this, but when I read things like : the projective plane, isn't actually a plane... I'm not getting happier...

Can I use that the one-point-compactification of ℝis homeomorphic to S¹?

I really like the activity on this forum... I am going to see if I can actually help someone...

Thanks again guy's!

still crying though... ;-)

 

[I like Serena]

Crying is the prerogative of the supreme onion, the saddener of worlds
Perhaps we should try to peel off a few of those projective layers.

 

[micromass]

What book are you using??

Did you ever hear of projective planes before??

 

[I like Serena]

I learned from the book Topologie by Klaus Jänich, but there's nothing in there about Projektiver Räume.

 

[Jooolz]

Yes I've heard about Pⁿ(ℝ), but I always try to avoid them... cause they're creeping me out...

Well, I use a book/reader that is made by a professor of my university...
And every now and then I skip through the pages of General Topology by Stephen Willard
Do you know it?

 

[Deveno]

the way i always thought about the projective plane is this:

start with a disk (you know, like a pancake, sort of). now if we sewed the edges of the disk "the right way", we'd get a sphere (imagine "closing up a basketball").

but if we "twisted" the edges, and sewed them up "backwards" (we'd need some extra dimensions to actually DO this, without the disk "intersecting itself"), we'd get the projective plane (it's weird, because it only has one side).

you can see a nifty animation of a polygonized model here (you have to imagine the self-intersections "aren't there"):

http://plus.maths.org/content/os/issue27/features/mathart/BoyUnfold