Difference between revisions of "0708-1300/Homework Assignment 3"

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{{In Preparation}}
 
  
 
==Reading==
 
==Reading==
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!on page(s)
 
!on page(s)
 
|- align=center
 
|- align=center
|1, 2, 3, S4, S5, S6
+
|S1, S2, 3, S4, S5
|82
+
|88
 
|- align=center
 
|- align=center
|1, 2, S3, S4, 5
+
|S1, 2, 3, S4, 5
|86
+
|89
 
|}
 
|}
 +
Note that these problems largely concern with material that we will not cover in class.
 +
 +
Problem 4 from page 88 needs a minor variation to be completely precise. See [[0708-1300/Errata to Bredon's Book|Brendon's book errata]] for more detail or much better find all the details your self.
  
 
==Due Date==
 
==Due Date==
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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>.
 +
* 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>.

Latest revision as of 10:56, 21 October 2007

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Contents

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)
S1, S2, 3, S4, S5 88
S1, 2, 3, S4, 5 89

Note that these problems largely concern with material that we will not cover in class.

Problem 4 from page 88 needs a minor variation to be completely precise. See Brendon's book errata for more detail or much better find all the details your self.

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, {\mathbb R}{\mathbb P}^2=S^2/(p=-p), inside {\mathbb R}^5. Do not do anything explicitly; just convince yourself that indeed you can find a small atlas (how small?), use it to embed {\mathbb R}{\mathbb P}^2 in some large {\mathbb R}^N (how large?), and figure out how many times you will need to use Sard's theorem before you're down to the target, {\mathbb R}^5.
  • Now see if you can come up with some cleverer way of viewing {\mathbb R}{\mathbb P}^2, that will allow you to explicitly embed it in {\mathbb R}^5.