0708-1300/Homework Assignment 3: Difference between revisions
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==Just for Fun== |
==Just for Fun== |
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* Trace the proof of the Whitney embedding theorem to find an embedding of the two dimensional real projective plane, <math>{\mathbb R}{\mathbb P}^2=S^2/(p=-p)</math>, inside <math>{\mathbb R}^5</math>. Do not do anything explicitly; just convince yourself that indeed you can find a small atlas (how small?), use it to embed <math>{\mathbb R}{\mathbb P}^2</math> in some large <math>{\mathbb R}^N</math> (how large?), and figure out how many times you will need to use Sard's theorem before you're down to the target, <math>{\mathbb R}^5</math>. |
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* Now see if you can come up with some cleverer way of viewing <math>{\mathbb R}{\mathbb P}^2</math>, that will allow you to explicitly embed it in <math>{\mathbb R}^5</math>. |
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Revision as of 08:55, 18 October 2007
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In Preparation
The information below is preliminary and cannot be trusted! (v)
Reading
Read sections 8-10 of chapter II of Bredon's book three times:
- First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
- Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
- And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.
Also, read section 12 of chapter I of Bredon's book, but you can be a little less careful here.
Doing
Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":
| problems | on page(s) |
|---|---|
| 1, 2, 3, S4, S5, S6 | 82 |
| 1, 2, S3, S4, 5 | 86 |
Due Date
This assignment is due in class on Thursday November 1, 2007.
Just for Fun
- Trace the proof of the Whitney embedding theorem to find an embedding of the two dimensional real projective plane, [math]\displaystyle{ {\mathbb R}{\mathbb P}^2=S^2/(p=-p) }[/math], inside [math]\displaystyle{ {\mathbb R}^5 }[/math]. Do not do anything explicitly; just convince yourself that indeed you can find a small atlas (how small?), use it to embed [math]\displaystyle{ {\mathbb R}{\mathbb P}^2 }[/math] in some large [math]\displaystyle{ {\mathbb R}^N }[/math] (how large?), and figure out how many times you will need to use Sard's theorem before you're down to the target, [math]\displaystyle{ {\mathbb R}^5 }[/math].
- Now see if you can come up with some cleverer way of viewing [math]\displaystyle{ {\mathbb R}{\mathbb P}^2 }[/math], that will allow you to explicitly embed it in [math]\displaystyle{ {\mathbb R}^5 }[/math].
