Difference between revisions of "0708-1300/Homework Assignment 5"

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* Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
 
* 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.
 
* 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.
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 +
'''Also,''' do the same with your own class notes - much of what we do for this part of the class is '''not''' in the textbook!
  
 
==Doing==
 
==Doing==
Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":
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Solve all of the following problems, but submit only your solutions of problems *,* and *:
{|align=center border=1 cellspacing=0 cellpadding=5
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|- align=center
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'''Problem 1.''' Let <math>M^n</math> be a manifold. Show that the following definitions for the orientability of <math>M</math> are equivalent:
!problems
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# There exists a nowhere vanishing <math>n</math>-form on <math>M</math>.
!on page(s)
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# There exists an atlas <math>\{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\}</math> for <math>M</math>, so that <math>\det(\phi_\alpha\phi^{-1}_\beta)>0</math> wherever that makes sense.
|- align=center
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|S1, S2
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'''Problem 2.''' Show that the tangent space <math>TM</math> of any manifold <math>M</math> is orientable.
|100-101
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|- align=center
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'''Problem 3.'''
|S1, S2, 3
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# Show that if <math>M</math> and <math>N</math> are orientable then so is <math>M\times N</math>.
|264
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# Show that if <math>M</math> and <math>M\times N</math> are orientable then so is <math>N</math>.
|}
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'''Problem 4.''' Show that <math>S^n</math> is always orientable.
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'''Problem 5.''' Recall that a form is called closed if it is in the kernel of <math>d</math> and exact if it is in the image of <math>d</math>. Show that every exact form is closed.
  
 
==Due Date==
 
==Due Date==

Revision as of 19:34, 21 November 2007

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In Preparation

The information below is preliminary and cannot be trusted! (v)

Contents

Reading

Read sections 1-3 of chapter V 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, do the same with your own class notes - much of what we do for this part of the class is not in the textbook!

Doing

Solve all of the following problems, but submit only your solutions of problems *,* and *:

Problem 1. Let M^n be a manifold. Show that the following definitions for the orientability of M are equivalent:

  1. There exists a nowhere vanishing n-form on M.
  2. There exists an atlas \{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\} for M, so that \det(\phi_\alpha\phi^{-1}_\beta)>0 wherever that makes sense.

Problem 2. Show that the tangent space TM of any manifold M is orientable.

Problem 3.

  1. Show that if M and N are orientable then so is M\times N.
  2. Show that if M and M\times N are orientable then so is N.

Problem 4. Show that S^n is always orientable.

Problem 5. Recall that a form is called closed if it is in the kernel of d and exact if it is in the image of d. Show that every exact form is closed.

Due Date

This assignment is due in class on Thursday December 6, 2007.

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