Geometry: Learn About Toruses & N-Balls

[Evilinside]

Right now, I am currently trying to study algebraic topology. In my geometrical studies I keep seeing objects such as toruses and n-balls. While an explanation of these objects is usually supplemented, I wanted to know if there was actually some subject or course out there from which I can learn about these objects specifically. Does anyone know of one?

 

[coalquay404]

Right now, I am currently trying to study algebraic topology. In my geometrical studies I keep seeing objects such as toruses and n-balls. While an explanation of these objects is usually supplemented, I wanted to know if there was actually some subject or course out there from which I can learn about these objects specifically. Does anyone know of one?

What you're talking about is shape, not necessarily geometry. If you're interested in n-tori (surfaces of genus n) or n-spheres, then you should consider trying to learn a bit about topology. Google for some introductory courses on topology and you should find what you're looking for.

 

[matt grime]

What else is there to know about them than the definition of what they are? Actually plenty, but you're studying algebraic topology to learn these facts about them. Beyond knowing what they are as (topologized) sets of points in R^n there is no other prerequisite knowledge required, I wouldn't have thought. Of course it'd be handy to understand that the torus and klein bottle are quotients of the plane for when you come to work out the fundamental groups.

 

[Evilinside]

I don't mean to sound incompetent, but I'm having difficulty studying Allen Hatcher's "Algebraic Topology" based his explanations of these objects alone. For example, A torus is the product of two S1 spheres. Algebraically that's simple, but intuitively and visually that makes no sense to me. How a doughnut comes from two circles in different planes, I don't know. Also why is a sphere always embedded in Rn+1 space and then it's boundary be an Sn-1 sphere? I've read that removing a single open disk from the 2-sphere gives a space which is homeomorphic to the closed 2-disk and removing two open disks from the 2-sphere gives a space homeomorphic to a closed cylinder. What is an open disk again and how does removing two from a 2 sphere give you a cylinder? Questions like this go on and on- mobius bands, klein bottles and other objects. I can still continue studying algebraic topology algebraically, computing fundamental groups, but I still feel like I'm in the dark and like I'm missing half of everything that is being said since I can never keep up with geometrical intiution the author expects you to know at that point.

I've studied point set topology before, but I will try look for a different source to see if it discusses these objects.

 

[matt grime]

It is really easy to explain if I were able to draw it fofr you. Indeed, you should draw a picture yourself, or get a bicycle inner tube, or even a donut, for what follows.

Take a donut draw a circle round the 'crown' (i.e. on the top of the donut, where the chocolate frosting is) going round the hole. There's one S^1, now pick a point on the circle as a base point. How do I specify the other points on the donut? I go round this circle some way and then travel on a second circle that goes off at right angles and loops round the body of the donut. There, the Torus T^1 is the same as S^1xS^1

The torus is also the same as the unit square with opposite sides glued together, which also shows that it is like S^1xS^1, since S^1 is the same as the unit interval with the end points identified.

Take S^2, remove an open disc, then you can imagine spreading it out by putting your fingers into the hole and pulling, and you'll get a disc, necessarily closed if we are only thinking of homeomorphism (homeomorphisms must be 'undoable' in this sense, homotopies need not be, thus a sphere with any point, open disc, or closed disc, removed is homotopic to a point.

An open disk is something homeomorphic to the set of points {(x,y) : x^2+y^2<1}, or better, it is well, a disc without its boundary (thus making it open).

Removing an open disc is punching a hole in the sphere, but leaving the boundary of the hole there (so what remains is closed).

These thigns really are a lot easier with a pad of paper to draw them on.

here, try this

http://www.answers.com/topic/torus

(googles third hit on searching for 'torus homeomorphic product circles', by the way), about a 1/3 of the way down it draws a picture of the torus and the two embedded circles idea.

 

[matt grime]

Also why is a sphere always embedded in Rn+1 space and then it's boundary be an Sn-1 sphere?

It isn't always so embedded, but that embedding is the easiest way to give a description of it. Spheres don't have boundaries, so I don't get the next part. The set of lines through the origin in complex two space is homeomorphic to a sphere, for instance.