Low dimensional Topology and Geometry – Dynamic Post

After posting “What worries me – the evolving list” it has occurred to me that I might have a number of dynamic posts.  Seems like an appropriate name for something that evolves with time (especially given the interest of some of those who write here).  A large part of me blogs to remedy my mistaken impression that I know anything.  These dynamic posts will be akin to files in a cabinet.  I’ll drop off finds here periodically and then create more lengthy posts off them.  Let the experiment begin.

I’ve well been aware of results in mathematics that seem to cluster around those dimensions that are dear to me: 3 and 4.  Over the past few months I have seen a number of clever persons talk about the same thing  (such as Atiyah) of which I wasn’t aware.  That was comforting at least. This doesn’t mean that interesting things don’t happen in more dimensions, they do.  In many, but finite, dimensions phase space gets really weird and statistical mechanical magic occurs. In infinitely many dimensions things become even more surreal (consider the unit ball in Banach spaces). This post however will keep track of low dimensional oddities and since I’ll be playing in these dimensions for a bit as I try and understand TQFTs it’ll be updated more often than not for a while.

(1) The cross product, as stolen from the Quaternions, works only in 3D for a number of reasons, one being that it’s the only space with a unique normal direction to a plane and the other is that 3C2=3C1 so there’s a way of associating, to each unit bivector a vector from \mathbb{R}^3.

(2) The only normed division algebras are $latex\mathbb{R}, \mathbb{C}, \mathbb{H}, \mbox{ and } \mathbb{O}$ this is due to Hurwitz’s theorem I believe and it corresponds to special things happening in dimensions 1, 2, 4, and 8.

(3) \mathfrak{so}(n)\approx\mathbb{R}^n only for n=3 (just saw this on TWF#3)

(4) S^3 is a double cover of SO(3).  This is cool because the 3-sphere is itself a Lie group and says something about its little brother.  I know this double covering also has a consequence in quantum but I can’t recall it.  Does this happen for other spheres? It’s also tied to the Quarternions (unit ones in particular) and so symmetries of the icosahedron and dodecahedron as well (Baez has stuff on this). See #5 for more

(5) Hopf fibration: S^1\hookrightarrow S^3\to S^2 which is related to a couple of the above items.  More importantly there are a limited number of spheres that fiber over other spheres, the ‘largest’ being 7,15, and 8.

(6) The Poincare Conjecture. If it looks like a sphere, smells like a sphere and is homotopy equivalent to a sphere it must be a sphere…right?  Well it depends, check out the Wikipedia page on the Generalized Poincare Conjecture. I would like to learn a lot more about this, basically it turned out easier to prove all the case for n>5 than for 3 and 4.  That’s odd no? This is related to the h-cobordism theorem which turned out to be easier to prove for n\geq

(7) In 2 dimensions there are a countable number of regular polygons.  In 3 dimensions there are five regular polyhedra, the Platonic solids.  In 4 dimensions there are six and in every dimension thereafter there are only three!

(8) If you look at the n-volume of the unit n-ball you’ll find that it increases for a while as dimensions increase but then peaks around dimension five and then tends to zero.  I’m cautious about this because the use of the word ‘volume’ is bound to bring some baggage with it.  What space a 18 dimensional sphere takes up isn’t the same kind of volume as the half gallon of milk in my refrigerator.  Though by contrast the n-volume of the unit n-cube is still one.

(9) I’m inclined to include knots here as well.  You can only tied knots (with S^1) in three dimensions.  An obvious retort is that you can knot surfaces in higher dimensions (in fact I’m excited to borrow a copy of “Knotted Surfaces and their Diagrams” by Scott J. Carter who has a number of other interesting titles.  This book was reviewed briefly by Baez in TWF 1.  The reason I include knots here is because there’s a ridiculous amount of physics going on with knots in the last couple of decades and it seems to concern knotted circles not surfaces.  If this correspondence plays out then once again there’s something special about n=3.

More to come…

 

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About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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