Time to get back to some topology, with the new computer. Short post this morning, but at least it's something. (I had a few posts queued up, just needing diagrams, but they got burned with the old computer. I had my work stuff backed up, but I don't let my personal stuff get into the company backups; I like to keep them clearly separated. And I didn't run my backups the way I should have for a few weeks.)
Last time, I started to explain a bit of patchwork: building manifolds from other manifolds using *gluing*. I'll have more to say about patchwork on manifolds, but first, I want to look at another way of building interesting manifolds.
At heart, I'm really an algebraist, and some of the really interesting manifolds can be defined algebraically in terms of topological product. You see, if you've got two manifolds **S** and **T**, then their product topology **S×T** is also a manifold. Since we already talked about topological product - both in classic topological terms, and in categorical terms, I'm not going to go back and repeat the definition. But I will just walk through a couple of examples of interesting manifolds that you can build using the product.
The easiest example is to just take some lines. Just a simple, basic line. That's a 1 dimensional manifold. What's the product of two lines? Hopefully, you can easily guess that: it's a plane. The standard cartesian metric spaces are all topological products of sets of lines: ℜn is the product of *n* lines.
To be a bit more interesting, take a circle - the basic, simple circle on a cartesian plane. Not the *contents* of the circle, but the closed line of the circle itself. In topological terms, that's a 1-sphere, and it's also a very simple manifold with no boundary. Now take a line, which is also a simple manifold.
What happens when you take the product of the line and the circle? You get a hollow cylinder.
What about if you take the product of the circle with *itself*? Thing about the definition of product: from any point *p* in the product **S×T**, you should be able to *project* an image of
**S** and an image of **T**. What's the shape where you can make that work right? The torus.
In fact the torus is a member of a family of topological spaces called the toroids. For any dimensionality *n*, there is an *n*-toroid which the the product of *n* circles. The 1-toroid is a circle; the 2-toroid is our familiar torus; the 3-toroid is a mess. (Beyond the 2-toroid, our ability to visualize them falls apart; what kind of figure can be *sliced* to produce a torus and a circle? The *concept* isn't too difficult, but the *image* is almost impossible.)
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What do you use for your diagramming?
A 3-torus isn't *that* hard to visualize: take the region bounded by two concentric 2-tori in 3-space, and imagine that its inside and outside boundaries are glued together.
I'll concede that things get hairy after that, though.
Micah:
Isn't that basically how most of the simple 4-d shapes are made? Take two concentric 3-d shapes, and glue the inner and outer boundaries of them. I'm not good at mental geometry, unfortunately, but I've tried quite hard to get a 4-cube in my head properly.
You can certainly make a lot of objects in 4-space by identifying boundaries. This technique, generally, is really useful.
Of course, it's just a specific example of the quotient topology. You may not be able to build everything this way, though. Your example of the 4-cube, which happens to be the product of four copies of the closed interval I, doesn't seem to be a quotient in any useful way.
Instead, I think the best way to think of it is just by using the cross product. If I^4 is the 4-cube, we can note that I^4=I^3xI, so the 4-cube is really just the 3-cube crossed by an interval.
You can also produce the 3-torus as a quotient of the 3-cube-- you just identify the faces in a particular way. This happens to be a model of the 3-torus that is helpful for doing certain calculations, like homology groups.
I used Google Images to see if anyone had tried to represent a 3-torus.
Some have, with interesting results on 4 pages.
Often they seem to confuse a genus-3 torus or variations of torus knots.
http://mathworld.wolfram.com/TripleTorus.html
and
http://mathworld.wolfram.com/TorusKnot.html
[MathWorld search for 3-torus yielded Torus {ring, horn, spindle}, Anosov map and Triple Torus {genus-3 torus}.]
http://mathworld.wolfram.com/search/index.cgi?q=3-torus
There are even 3 pages for the 4-torus at Google Images.
The most interesting representation may be 'Volume Visualization II' at The Laboratory for Engineering Man/Machine Systems (LEMS) of Brown University.
There is a section labeled:
Part A: Saddle Points
11) Torus Normals at Saddle-Points: View 1
12) Torus Normals at Saddle-Points: View 2
13) Torus Normals at Reduced Saddle-Points: View 1 (See discussion)
14) Torus Normals at Reduced Saddle-Points: View 2 (See discussion)
15) Torus Maximum Curvatures at Saddle-Points
16) Torus Minimum Curvatures at Saddle-Points
17) Bone Normals at Saddle-Points:
18) Bone Maximum Curvatures at Saddle-Points
19) Bone Minimum Curvatures at Saddle-Points
with illustrations in 'A. Differential Properties at Saddle Points'.
http://www.lems.brown.edu/~msj/cs292/assign4/volumeVis.html
Saddle Points in mathematical game theory are potential Nash Equilibria [U-RI].
"... equilibrium decision point is often termed a saddle point. Such a point is often also known as a minimax point [von Neumann] ..."
http://www.math.uri.edu/~kulenm/mth381pr/GAMETH/gametheory.html
Plasma tori associated with planetary magnetospheres are likely ubiquitous such as this illustration [Figure 11.13] of Jupiter [from Nordic Institute for Theoretical Physics].
http://www.nordita.dk/~steen/fysik51/ast/ast116_files/AT41104.HTM
Are tori saddle points a link between game theory and representation [Lie] groups / algebras?
I was expecting you to move on this way:
"In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure...."
http://en.wikipedia.org/wiki/Trivial_bundle
Do you have any comment on "Finite Generation of the Canonical Ring"
http://www.math.columbia.edu/~woit/wordpress/?p=481#comments
"The last few weeks have seen the appearance of two papers giving very different proofs of a quite important result in algebraic geometry, resolving a question that had been open for a very long time, and in the process helping to make progress in the classification of higher dimensional projective algebraic varieties. Readers should be warned that this doesn't have anything to do with physics, and my knowledge of this kind of mathematics is highly shaky, so I'm relying largely on second-hand information from people much better informed than myself...."